Fluid-saturated Porous Medium Research Articles

Onset of convective instability in an inclined porous medium

The diffusion of a solute from a concentrated source into a horizontal, stationary, fluid-saturated porous medium can lead to a convective motion when a gravitationally unstable density stratification evolves. In an inclined porous medium, the convective flow becomes intricate as it originates from a combination of diffusion and lateral flow, which is dominant near the source of the solute. Here, we investigate the role of inclination on the onset of convective instability by linear stability analyses of Darcy's law and mass conservation for the flow and the concentration field. We find that the onset time increases with the angle of inclination (θ) until it reaches a cutoff angle beyond which the system remains stable. The cutoff angle increases with the Rayleigh number, Ra. The evolving wavenumber at the onset exhibits a lateral velocity that depends non-monotonically on θ and linearly on Ra. Instabilities are observed in gravitationally stable configurations (θ≥90°) solely due to the nonuniform base flow generating a velocity shear commonly associated with Kelvin–Helmholtz instability. These results quantify the role of medium tilt on convective instabilities, which is of great importance to geological CO2 sequestration.

A Macro-microscopic Coupled Constitutive Model for Fluid-Saturated Porous Media with Compressible Constituents

The paper provides a macro-microscopic coupled constitutive model for fluid-saturated porous media with respect to the compressibility of the solid skeleton, the real solid material and the fluid phase. The derivation of the model is carried out based on the porous media theory and is consistent with the second law of thermodynamics. In the present paper, two different sets of independent variables are introduced to implement the coupled behavior between the compressibility of the solid skeleton and the real solid material. Altogether the proposed model exploits five independent variables, i.e. the deviatoric part of the right Cauchy-Green deformation tensor, the partial density of solid phase, the density of the real solid material, the density of the real fluid material and the relative velocity of the fluid phase. Subsequently, the linearized version of the proposed constitutive model is also presented and compared with some models by other authors. It is found that Biot's model can also be derived based on the linearized version of the proposed model, which indicates that the present work bridges the gap between the porous media theory and Biot's model. Compared with Biot's model, the present model can provide the evolution of the porosity by considering the volumetric strain of the solid skeleton and the volumetric strain of the real solid material.

Dynamic soil consolidation model using a nonlocal continuum poroelastic damage approach

AbstractWe present a novel dynamic poroelastic model for soil consolidation in an isotropic homogeneous fluid saturated porous media. A nonlocal integral‐type continuum damage formulation is applied to describe the damage evolution under dynamic excitations, in which a bilinear damage law is assumed. The governing equations are obtained by considering the conservation of momentum and mass balance for the solid–fluid mixtures, in which the fluid flow obeys Darcy's seepage law in the entire domain. The numerical solution of the fully coupled problem is achieved via a mixed FEM displacement–pressure element formulation. The mechanical equilibrium equations are evolved in time using a Newmark method and the resulting nonlinear system is solved via a Newton–Raphson method. Consistent linearization is then used to obtain the block Jacobian matrix analytically and the linear system is solved monolithically at each time step. Several numerical results are presented to study soil consolidation problems under harmonic excitations. We investigate the time‐dependence response of the skeleton displacement, pressure, and damage evolution. In addition, the examples demonstrate that the nonlocal integral‐type damage model can effectively overcome mesh dependence and yield smooth behavior.

Semi-analytical finite element method for simulating chemical dissolution-front instability problems in fluid-saturated porous media

PurposeThe objective of this paper is to develop a semi-analytical finite element method for solving chemical dissolution-front instability problems in fluid-saturated porous media.Design/methodology/approachThe porosity, horizontal and vertical components of the pore-fluid velocity and solute concentration are selected as four fundamental unknown variables for describing chemical dissolution-front instability problems in fluid-saturated porous media. To avoid the use of numerical integration, analytical solutions for the property matrices of a rectangular element are precisely derived in a purely mathematical manner. This means that the proposed finite element method is a kind of semi-analytical method. The column pivot element solver is used to solve the resulting finite element equations of the chemical dissolution-front instability problem.FindingsThe direct use of horizontal and vertical components of the pore-fluid velocity as fundamental unknown variables can improve the accuracy of the related numerical solution. The column pivot element solver is useful for solving the finite element equations of a chemical dissolution-front instability problem. The proposed semi-analytical finite element method can produce highly accurate numerical solutions for simulating chemical dissolution-front instability problems in fluid-saturated porous media.Originality/valueAnalytical solutions for the property matrices of a rectangular element are precisely derived for solving chemical dissolution-front instability problems in fluid-saturated porous media. The proposed semi-analytical finite element method provides a useful way for understanding the underlying dynamic mechanisms of the washing land method involved in the contaminated land remediation.

Entropy generation for MHD natural convection in enclosure with a micropolar fluid saturated porous medium with Al2O3Cu water hybrid nanofluid

This contribution gives a numerical investigation of buoyancy-driven flow of natural convection heat transfer and entropy generation of non-Newtonian hybrid nanofluid (Al2O3-Cu) within an enclosure square porous cavity. Hybrid nanofluids represent a novel type of enhanced active fluids. During the current theoretical investigation, an actual available empirical data for both thermal conductivity and dynamic viscosity of hybrid nanofluids are applied directly. Numerical simulation have been implemented for solid nanoparticles, the volumetric concentration of which varies from 0.0% (i.e., pure fluid) to 0.1% of hybrid nanofluids. Heat and sink sources are situated on a part of the left and right sides of the cavity with length B, while the upper and bottom horizontal sides are kept adiabatic. The stated partial differential equations describing the flow are mutated to a dimensionless formulas, then solved numerically via the help of an implicit finite difference approach. The acquired computations are given in terms of streamlines, isotherms, isomicrorotations, isoconcentraions, local Began number, total entropy, local and mean Nusselt numbers. The data illustrates that variations of ratio of the average Nusselt number to the averageNusselt of pure fluid Num+ is a decreasing function of Ha and φ, while e+ is an increasing function of Ha and φ parameters of hybrid nanofluid.

Transitions and bifurcations in couple stress fluid saturated porous media using a thermal non-equilibrium model

In this article, we study the stability and transition of couple stress fluid saturated porous media, heated from below and cooled from above by employing a thermal non-equilibrium model. Careful analysis shows that the thermal non-equilibrium model has a global attractor, and the global attractor only consists of the basic solution if the Rayleigh number is equal or below a threshold. In generic case where the transitioning eigenvalue has multiplicity one, we show that the transition involved is of continuous type, and the basic solution will be bifurcated to two stable convection solutions which attract globally. If the leading eigenvalue multiplicity is two, the transition is also continuous and a global attractor homeomorphic to the unit circle bifurcates. The attractor then contains four steady-state convection solutions, two of which are stable while the other two are unstable. Further numerical works give more details in the transition process including the bifurcated roll structure.

Transient Three-Dimensional Natural Convection in Confined Porous Media with Time-Periodic Boundary Conditions

Transient three-dimensional natural convection in confined fluid-saturated porous media had been investigated numerically through this work. The geometry selected is a box with time-periodic temperature variation at the vertical sides and constant wall temperature at the top and bottom. In this investigation, the momentum equation of flow through fluid-saturated porous media had been transformed to a vector potential form and solved numerically using the Successive Over Relaxation method while the energy equation is solved using the three-dimensional Alternating Direction Implicit method. The values of the Rayleigh number under investigation are (150, 200, 250 and 300). The results are presented in a form of contour maps for the temperature and a velocity vector maps for the velocity. The results reveal that the temperature within the box is increased as the time or Rayleigh number increase.The Nusselt number varies inversely with the time and directly with the Rayleigh number. Two cell convective patterns are obtained in the y-z

Dual solution for double-diffusive mixed convection opposing flow through a vertical cylinder saturated in a Darcy porous media containing gyrotactic microorganisms

The steady mixed convection flow towards an isothermal permeable vertical cylinder nested in a fluid-saturated porous medium is studied. The Darcy model is applied to observe bioconvection through porous media. The suspension of gyrotactic microorganisms is considered for various applications in bioconvection. Appropriate similarity variables are opted to attain the dimensionless form of governing equations. The resulting momentum, energy, concentration, and motile microorganism density equations are then solved numerically. The resulting dual solutions are graphically visualized and physically analyzed. The results indicate that depending on the systems' parameters, dual solutions exist in opposing flow beyond a critical point where both solutions are connected. Our results were also compared with existing literature.

Effects of porosity, orientation and connectivity of microcracks on dispersion and attenuation of fluid-saturated rocks using an upscaling numerical modelling of the squirt flow mechanism

Squirt flow is a wave-induced fluid flow mechanism to account for the velocity dispersion and attenuation of fluid-saturated porous media. Theoretical models of squirt flow cannot deal with complex microcrack-pore networks and thus cannot give accurate dispersion and attenuation curves with respect to frequency. We adopt a numerical oscillatory compressibility test method, based on the quasi-static poroelastic equation of Biot, to model the squirt flow and calculate corresponding P-wave modulus dispersion and attenuation. We also designed nine porous medium models containing different crack-pore configurations and investigated the effects of microcrack porosity, orientation and connectivity on the elastic responses. Modelling results show that not only microcrack porosity but also their orientation has control on the magnitude of the P-wave modulus dispersion and attenuation. Preferentially aligned microcracks may cause frequency-dependent anisotropy in the P-wave modulus and attenuation. Microcrack connectivity has a negligible influence on the dispersion and attenuation magnitude but a large influence on the characteristic frequency of the squirt flow, which is not predicted by theoretical models based on simple representative element volume of crack-pore structure. Therefore, detailed geometry of crack-pore network, of which microcrack porosity, orientation, aspect ratio and connectivity are vital factors, dictates the unique variations in the elastic modulus and associated attenuation with frequency caused by the squirt flow. Pore structure obtained by CT scanning representative of that of the whole rock is needed to obtain more accurate poroelastic responses of the rock with the numerical oscillatory compressibility test method. In this regard, we provide a powerful tool for evaluating dispersion and attenuation of fluid-saturated rock media with complex crack-pore structure, which have potential applications in seismic exploration of hydrocarbon in the subsurface.

Explicit meshfree {{varvec{u}}}-{{varvec{p}}}_mathbf{mathrm{w}} solution of the dynamic Biot formulation at large strain

In this paper, an efficient and robust methodology to simulate saturated soils subjected to low-medium frequency dynamic loadings under large deformation regime is presented. The coupling between solid and fluid phases is solved through the dynamic reduced formulation u-p_mathrm{w} (solid displacement – pore water pressure) of the Biot’s equations. The additional novelty lies in the employment of an explicit two-steps Newmark predictor-corrector time integration scheme that enables accurate solutions of related geomechanical problems at large strain without the usually high computational cost associated with the implicit counterparts. Shape functions based on the elegant Local Maximum Entropy approach, through the Optimal Transportation Meshfree framework, are considered to solve numerically different dynamic problems in fluid saturated porous media.

Lattice Boltzmann Simulation for Flow Inside Open-Ended Porous Medium With Partially Thermally Active Walls

Abstract Free convection heat transfer and flow characteristics in an open-ended enclosure occupied with fluid saturated porous media are highlighted in this paper. All numerical investigations are achieved using the mesoscopic approach thermal Lattice Boltzmann Method (LBM) by using the Darcy-Forchheimer model. The bottom and the top sides of the porous enclosure is thermally isolated with complete or partially heated vertical wall facing the opening sidewall. The partial slice of left wall of the enclosure with a fixed heating length as (H/3) is isothermally heated at the middle, top and bottom locations. However, right side is open to the ambient physical conditions. The influences of partial heating location on free convection characteristics, namely, isotherms, streamlines, centerline variations of horizontal and vertical, average and local Nusselt numbers are explored for Darcy number of 0.01, porosity of 0.4, Rayleigh number of 106 and unity Prandtl number.

Development of a 3D fluid-saturated element for dynamic analysis of two-phase media in ABAQUS based on u-U formed equations

The commonly used finite element software does not provide a three-dimensional (3D) fluid-saturated element available for dynamic analysis of two-phase media, which extremely limits their advantages in solving complicated engineering problems. This paper develops a user element for 3D dynamic analysis of fluid-saturated porous media in ABAQUS based on the u-U formed equations. The dynamic governing formulations are firstly revisited, and an implicit time integration algorithm is adopted to solve the dynamic equations. Then, the major development steps, the structure of the subroutine file, and the keywords directly related to the user element in the input file are carefully presented for the application of the user element. Moreover, the equivalent linear approach is introduced with a detailed realization procedure to consider the soil nonlinearity. Finally, the effectiveness and accuracy of the developed user element are validated by comparing the results with those from four different analytical and numerical solutions. The proposed numerical method with user element has a great potential in modeling practical engineering problems due to the prominent advantages of ABAQUS in simulating complicated materials and irregular geometrics.

On the seismic response of a periodic sequence of three thin layers saturated by two-phase fluids

The conversion of fast to slow diffusion P waves in fluid-saturated porous media induces attenuation and dispersion of waves at seismic frequencies. This effect, known as wave-induced fluid flow, occurs at mesoscopic scales, which are much larger than the average pore size and much smaller than the average fast P-wave wavelength. When analyzing this mechanism in hydrocarbon reservoirs with the pore space saturated by multiphase fluids, it is important to include capillary pressure effects and flow interaction between fluids, which cause additional attenuation and velocity dispersion of P waves. We have developed a procedure to determine the phase velocities and dissipation factors in a medium composed of a periodic sequence of three poroelastic thin layers saturated by two-phase fluids. The methodology consists of applying compressibility tests to representative samples of the material, which are defined as boundary-value problems solved using a finite-element procedure. First, we analyze the case of two-phase fluid saturation on each layer, and the results are compared with those of the single-phase (effective) fluid case. Then, several cases of patchy saturation are presented, indicating that residual and wetting fluid saturation play an important role in determining the P-wave velocities and dissipation factors.

Reflection and transmission phenomena of SH waves in fluid saturated porous medium with corrugated interface

This article deals with the behavior of reflection/transmission phenomena on the propagation of SH waves in a corrugated fluid-saturated porous layer sandwiched between heterogeneous viscoelastic and orthotropic half-spaces. Various mathematical tools such as Rayleigh’s method of approximation, Snell’s law, Crammer’s law are applied to get the analytical expression for the reflection/transmission coefficient and energy ratio for each scattered wave. A numerical treatment has been performed for a particular model to study the nature of various affecting parameters such as heterogeneity, viscosity, porosity, angle of incidence, and wave number. A comparative study has been made for the graphical observation to show the effect of the various affecting parameters in the presence and absence of porosity. The reflection/transmission coefficient, energy ratio for each wave (regularly and irregularly reflected/transmitted wave) are calculated analytically and given a graphical representation for a particular model. It has been shown that the conservation of energy rule holds very well in the model. The finding of this article may be valuable in the field of seismology and geophysics to enhance the knowledge about the structure of rock in the interior of the Earth and its elastic property.

A hybrid Laplace‐Galerkin method for thermo‐hydro‐mechanical coupling in fluid saturated porous media: Application for borehole problems

AbstractA hybrid Laplace‐Galerkin method is proposed for modeling thermo‐hydro‐mechanical coupled problems of fluid saturated porous media in unsteady state regime. It consists of transforming the weak form equations into the Laplace space where they are solved by the Galerkin finite element method. Results in time space are then obtained by the Stehfest method of inverse Laplace transformation. This technique does not require any time stepping comparing to the traditional implicit or explicit time discretization method. Its exactitude is confirmed by comparing with existent analytical results as well as the one obtained by the implicit time stepping method.

Influence of interface condition on reflection of elastic waves in fluid-saturated porous media

The characteristics of P-wave reflection coefficients can be dependent on the properties of the interface separating the two dissimilar poroelastic media. To evaluate the influence of the interface condition, we have developed a general theoretical formulation of seismic reflection and transmission of waves at arbitrary incidence angles in fluid-saturated porous media. The formulation is derived based on the quasistatic Biot’s theory and incorporates a more general boundary condition for fluid pressure. Simple analytic expressions are obtained for reflection and transmission coefficients at normal incidence. The equations incorporate an interfacial impedance that can be used to effectively characterize the interface condition. Based on the new formulation, we study the coupled effects of dynamic fluid flow and the interface condition on the reflection and transmission coefficients. Two pertinent reflection scenarios in exploration geophysics are considered in the numerical analysis, i.e., a gas-water contact and a free fluid overlying a gas-saturated medium. We examine the corresponding P-wave reflection coefficient resulting from different interface conditions such as imperfect hydraulic contact, capillary pressure, and open- and sealed-pore interfaces. Our results reveal that the interface condition can affect the frequency dependence and amplitude-versus-angle signatures of the reflection coefficient.

Effect of stresses on wave propagation in fluid-saturated porous media

The mechanism of the effect of stresses on wave propagation in a fluid-saturated porous media has not been well understood. The goal of this paper is to fill this gap. First we formulated the general equations of motion in a homogeneously pre-stressed fluid-saturated medium and use them to derive explicit expressions of velocity dependence on stresses. Equations for fast and slow longitudinal waves allow substantial simplifications. The feasibility of fast longitudinal wave simplification is verified by comparison with experimental data available in literature.

Effect of local thermal non-equilibrium on the onset of convection in an anisotropic bidispersive porous layer

The onset of thermal convection of a fluid saturated anisotropic bidisperse porous medium under the condition of local thermal non-equilibrium is investigated. We have studied the case of flow in the macropores and micropores when the porous materials are of Darcy type. The temperatures in the macropores and micropores are allowed to be different. We concentrate our attention on the state of a permeability tensor is transversely isotropic with the isotropy axis in the vertical direction of gravity and the permeability ratios of vertical to horizontal are different in the macropores and micropores . The effect of various parameters on the stationary convection is discussed. In particular the effects of macro permeabilities, the micro permeabilities, the measure between the permeability in the macro phase and micro phase and various interaction parameters on the stationary convection are studied. The numerical results are presented for free-free boundary conditions.

Unified approach on numerical implementation of heatfunction boundary conditions for accurate prediction of heatlines involving steady convection in enclosures

The tool “Heatlines” is useful to understand heat flow and thermal management in enclosures experiencing convection. The heatline concept was first introduced by Bejan in the early 80’s. This article starts with the fundamental principles of heatlines or heat flow trajectories followed by a generalized governing equation for heatfunction (Π) applicable for fluid, nanofluid, or fluid-saturated porous medium experiencing steady natural or mixed convection within enclosures. An unified approach on derivation of heatfunction boundary conditions is presented based on Dirichlet conditions at junction of walls while Neumann conditions at walls for Π. The solution of Π also requires the selection of location of Π = 0 and this article outlines a generalized overview on the invariance of heatline trajectories irrespective of the location of Π = 0 for enclosures with or without adiabatic walls. Open loop or end to end heatlines demonstrate the heat flow path for conductive heat transport whereas open loop heatlines cross various regimes of dominant conduction or convection zones connecting the hot walls to the cold walls for convective heat transport. Closed loop heatline vortices (qualitatively similar to the flow vortices) depict the convective heat flow via the trajectories of the fluid flow vortices. A few feasible heatlines for conduction and convection dominant heat transport in enclosures (steady state) have been presented to illustrate the physics of “heatlines.” This article also addresses a few earlier works depicting discrepancies in heatline trajectories [discrepancies: heat flow to and from the same walls (isothermally hot/cold or adiabatic) or heat flow from the hot/cold to the adiabatic wall(s)] and proposes qualitative pictures for feasible heatlines for a few earlier case studies. Understanding the feasible implementation of heatlines is required as the inaccurate representation of heatlines may mislead future researchers in interpreting the energy flow.

Effects of mathematical transforms on theoretical analysis and computational simulation of chemical dissolution-front instability within fluid-saturated porous media

Reactive mass transport, in which chemical dissolution front may become unstable, is a common phenomenon in the field of groundwater hydrology. The use of mathematical transforms can convert many scientific and engineering problems from the conventional time–space domain into a generalized time–space domain, so that analytical solutions, which are impossible to be directly obtained in the conventional time–space domain, can be derived in the generalized time–space domain. The theoretical analysis and computational simulation of dissolution-timescale chemical dissolution-front instability within fluid-saturated porous media is no exception. To investigate how different mathematical transforms can affect the theoretical analyses and computational simulation of chemical dissolution-front instability within fluid-saturated porous media, two different approaches are considered to select mathematical transforms in this study. In the first approach, the mathematical transform mainly consists of a much larger timescale than the dissolution timescale and a length-scale that is independent of the mineral dissolution ratio (MDR), while in the second approach, the mathematical transform mainly consists of the dissolution timescale and the MDR-dependent length-scale. The related theoretical and computational results have demonstrated that: (1) the use of two different mathematical transforms has no effect on the analytical solution to the dispersion equation of a chemical dissolution-front instability problem in the original physical time–space domain. (2) The use of different mathematical transforms may have significant effects on computationally simulating the evolution patterns of unstable chemical dissolution-front in finite space domains, which are filled with fluid-saturated porous media. (3) If the mathematical transform is appropriately selected, then the dissolution-timescale chemical dissolution-front instability problem in fluid-saturated porous media is mathematically solvable.