Pore Scale Modeling of Reactive Transport Involved in Geologic CO<Subscript>2</Subscript> Sequestration

We apply a multi-component reactive transport lattice Boltzmann model developed in previous studies for modeling the injection of a CO₂-saturated brine into various porous media structures at temperatures T = 25 and 80°C. In the various cases considered the porous medium consists initially of calcite with varying grain size and shape. A chemical system consisting of Na⁺, Ca²⁺, Mg²⁺, H⁺, CO₂^°(aq){{\rm CO}_2^{\circ}{\rm (aq)}}, and Cl⁻ is considered. Flow and transport by advection and diffusion of aqueous species, combined with homogeneous reactions occurring in the bulk fluid, as well as the dissolution of calcite and precipitation of dolomite are simulated at the pore scale. The effects of the structure of the porous media on reactive transport are investigated. The results are compared with a continuum-scale model and the discrepancies between the pore- and continuum-scale models are discussed. This study sheds some light on the fundamental physics occurring at the pore scale for reactive transport involved in geologic CO₂ sequestration.     

Jump Condition for Diffusive and Convective Mass Transfer Between a Porous Medium and a Fluid Involving Adsorption and Chemical Reaction

In this paper, mass transfer at the fluid–porous medium boundaries is studied. The problem considers both diffusive and convective transport, along with adsorption and reaction effects in the porous medium. The result is a mass flux jump condition that is expressed in terms of effective transport coefficients. Such coefficients (a total dispersion tensor and effective reaction and adsorption coefficients) may be computed from the solution of the corresponding closure problem here stated and solved as a function of the Péclet number (Pe), the porosity and a local Thiele modulus. For the case of negligible convective transport (i.e., ), the closure problem reduces to the one recently solved by the authors for diffusion and reaction between a fluid and a microporous medium.     

Thermophoresis particle deposition in a non-Darcy porous medium under the influence of Soret, Dufour effects

Thermophoresis particle deposition in free convection on a vertical plate embedded in a fluid saturated non-Darcy porous medium is studied using similarity solution technique. The effect of Soret and Dufour parameters on concentration distribution, wall thermophoretic deposition velocity, heat transfer and mass transfer is discussed in detail for different values of dispersion parameters (Ra _γ, Ra _ξ) inertial parameter F and Lewis number Le. The result indicates that the Soret effect is more influential in increasing the concentration distribution in both aiding as well as opposing buoyancies. Also, the non-dimensional heat transfer coefficient and non-dimensional mass transfer coefficient changes according to different values of thermophoretic coefficient k.     

Influence of lateral mass flux on mixed convection heat and mass transfer over inclined surfaces in porous media

 The effect of lateral mass flux on mixed convection heat and mass transfer in a saturated porous medium adjacent to an inclined permeable surface is analyzed. A similarity solution is obtained when surface temperature and concentration, free stream velocity and injection/suction velocity of fluid are prescribed as power functions of distance from the leading edge. The cases when the flow and buoyancy forces are in the same and opposite directions are discussed both for aiding and opposing buoyancy effects. The governing parameters are the mixed convection parameter Gr, the Lewis number Le, the buoyancy ratio N, the lateral mass flux parameter f _w, representing the effects of injection or withdrawal of fluid at the wall, and λ which specifies three cases of the inclined plate. The interactive effect of these parameters on heat and mass transfer rates are presented. It is observed that the diffusion ratio (Le) has a more pronounced effect on concentration field than on flow and temperature fields. It is found that the rates of heat and mass transfer increase with suction and decrease with injection of the fluid. Received on 31 August 2000 / Published online: 29 November 2001     

Non-Darcy Free Convection of Power-Law Fluids Over a Two-Dimensional Body Embedded in a Porous Medium

A boundary layer analysis was presented to study the non-Darcy-free convection of a power-law fluid over a non-isothermal two-dimensional body embedded in a porous medium. The Ostwald-de Waele power-law model was used to characterize the non-Newtonian fluid behavior. Similarity solutions were obtained with variations in surface temperature or surface heat flux. In view of the fact that most of the non-Newtonian fluids have large Prandtl numbers, this study was directed toward such fluids. The effects of the porous medium parameters, k ₁ and k ₂, body shape parameter, m, and surface thermal variations parameter, p, as well as the power-law index, n, were examined.     

Thermal radiation effects on MHD flow past a semi-infinite inclined plate in the presence of mass diffusion

MHD mixed free-forced heat and mass convective steady incompressible laminar boundary layer flow of a gray optically thick electrically conducting viscous fluid past a semi-infinite inclined plate for high temperature and concentration differences is studied. A uniform magnetic field is applied perpendicular to the plate. The density of the fluid is assumed to reduce exponentially with temperature and concentration. The usual Boussinesq approximation is neglected due to the high temperature and concentration differences between the plate and the ambient fluid. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The boundary layer equations governing the flow are reduced to ordinary differential equations, which are numerically solved by applying an efficient technique. The effects of the density/temperature parameter n, the density/concentration parameter m, the local magnetic parameter M_x and the radiation parameter R are examined on the velocity, temperature and concentration distributions as well as the coefficients of skin-friction, heat flux and mass flux.     

The onset of buoyancy-driven convection in a ferromagnetic fluid saturated porous medium

The onset of buoyancy-driven convection in an initially quiescent ferrofluid saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated. The Brinkman-Lapwood extended Darcy equation with fluid viscosity different from effective viscosity is used to describe the flow in the porous medium. The lower boundary of the porous layer is assumed to be rigid-paramagnetic, while the upper paramagnetic boundary is considered to be either rigid or stress-free. The thermal conditions include fixed heat flux at the lower boundary, and a general convective–radiative exchange at the upper boundary, which encompasses fixed temperature and fixed heat flux as particular cases. The resulting eigenvalue problem is solved numerically using the Galerkin technique. It is found that increase in the Biot number Bi, porous parameter σ, viscosity ratio Λ, magnetic susceptibility χ, and decrease in the magnetic number M ₁ and non-linearity of magnetization M ₃ is to delay the onset of ferroconvection in a porous medium. Further, increase in M ₁, M ₃, and decrease in χ, Λ, σ and Bi is to decrease the size of convection cells.     

Non-similar Solution for Natural Convective Boundary Layer Flow Over a Sphere Embedded in a Porous Medium Saturated with a Nanofluid

A boundary layer analysis is presented for the natural convection past an isothermal sphere in a Darcy porous medium saturated with a nanofluid. Numerical results for friction factor, surface heat transfer rate, and mass transfer rate have been presented for parametric variations of the buoyancy ratio parameter N _r, Brownian motion parameter N _b, thermophoresis parameter N _t, and Lewis number L _e. The dependency of the friction factor, surface heat transfer rate (Nusselt number), and mass transfer rate (Sherwood number) on these parameters has been discussed.     

Turbulence Modeling Effects on the Prediction of Equilibrium States of Buoyant Shear Flows

The effects of turbulence modeling on the prediction of equilibrium states of turbulent buoyant shear flows were investigated. The velocity field models used include a two-equation closure, a Reynolds-stress closure assuming two different pressure-strain models and three different dissipation rate tensor models. As for the thermal field closure models, two different pressure-scrambling models and nine different temperature variance dissipation rate ɛ_τ) equations were considered. The emphasis of this paper is focused on the effects of the ɛ_τ-equation, of the dissipation rate models, of the pressure-strain models and of the pressure-scrambling models on the prediction of the approach to equilibrium turbulence. Equilibrium turbulence is defined by the time rate of change of the scaled Reynolds stress anisotropic tensor and heat flux vector becoming zero. These conditions lead to the equilibrium state parameters, given by /ɛ, _τ/ɛ_τ, , Sk/ɛ and G/ɛ, becoming constant. Here, and _τ are the production of turbulent kinetic energy k and temperature variance , respectively, ɛ and ɛ_τ are their respective dissipation rates, R is the mixed time scale ratio, G is the buoyant production of k and S is the mean shear gradient. Calculations show that the ɛ_τ-equation has a significant effect on the prediction of the approach to equilibrium turbulence. For a particular ɛ_τ-equation, all velocity closure models considered give an equilibrium state if anisotropic dissipation is accounted for in one form or another in the dissipation rate tensor or in the ɛ-equation. It is further found that the models considered for the pressure-strain tensor and the pressure-scrambling vector have little or no effect on the prediction of the approach to equilibrium turbulence. Received 21 April 2000 and accepted 21 February 2001     

Natural Convection in a Cavity Filled with Porous Layers on the Top and Bottom Walls

Natural convection in a partially filled porous square cavity is numerically investigated using SIMPLEC method. The Brinkman-Forchheimer extended model was used to govern the flow in the porous medium region. At the porous-fluid interface, the flow boundary condition imposed is a shear stress jump, which includes both the viscous and inertial effects, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The results are presented with flow configurations and isotherms, local and average Nusselt number along the cold wall for different Darcy numbers from 10⁻¹ to 10⁻⁶, porosity values from 0.2 to 0.8, Rayleigh numbers from 10³ to 10⁷, and the ratio of porous layer thickness to cavity height from 0 to 0.50. The flow pattern inside the cavity is affected with these parameters and hence the local and global heat transfer. A modified Darcy–Rayleigh number is proposed for the heat convection intensity in porous/fluid filled domains. When its value is less than unit, global heat transfer keeps unchanged. The interfacial stress jump coefficients β ₁ and β ₂ were varied from  −1 to +1, and their effects on the local and average Nusselt numbers, velocity and temperature profiles in the mid-width of the cavity are investigated.     

Fully Developed Forced Convection in a Parallel Plate Channel with a Centered Porous Layer

In this study, fully developed heat and fluid flow in a parallel plate channel partially filled with porous layer is analyzed both analytically and numerically. The porous layer is located at the center of the channel and uniform heat flux is applied at the walls. The heat and fluid flow equations for clear fluid and porous regions are separately solved. Continues shear stress and heat flux conditions at the interface are used to determine the interface velocity and temperature. The velocity and temperature profiles in the channel for different values of Darcy number, thermal conductivity ratio, and porous layer thickness are plotted and discussed. The values of Nusselt number and friction factor of a fully clear fluid channel (Nu _cl = 4.12 and fRe _cl = 24) are used to define heat transfer increment ratio (e_th = Nu_p/Nu_cl)({\varepsilon _{\rm th} =Nu_{\rm p}/Nu_{\rm cl})} and pressure drop increment ratio (e_p = fRe_p/fRe_cl )({\varepsilon_{\rm p} =fRe_{\rm p}/fRe_{\rm cl} )} and observe the effects of an inserted porous layer on the increase of heat transfer and pressure drop. The heat transfer and pressure drop increment ratios are used to define an overall performance (e = e_th/e_p)({\varepsilon = \varepsilon_{\rm th}/\varepsilon_{\rm p})} to evaluate overall benefits of an inserted porous layer in a parallel plate channel. The obtained results showed that for a partially porous filled channel, the value of e{\varepsilon} is highly influenced from Darcy number, but it is not affected from thermal conductivity ratio (k _r) when k _r > 2. For a fully porous material filled channel, the value of e{\varepsilon} is considerably affected from thermal conductivity ratio as the porous medium is in contact with the channel walls.     

An Extended-FEM Model for CO2 Leakage Through a Naturally Fractured Cap-Rock During Carbon Dioxide Sequestration

In this paper, a numerical model is developed for the assessment of carbon dioxide transport through naturally fractured cap-rocks during CO₂ sequestration in underground aquifers. The cap-rock contains two types of fracture with different length scales: micro-cracks (fissures) and macro-cracks (faults). The effect of micro-cracks is incorporated implicitly by modifying the intrinsic permeability tensor of porous matrix, while the macro-cracks are modeled explicitly using the extended finite element method (X-FEM). The fractured porous medium is decomposed into the porous matrix and fracture domain, which are occupied with two immiscible fluid phases, water and CO₂. The flow inside the matrix domain is governed by the Darcy law, while the flow within the fracture is modeled using the Poiseuille flow. The mass conservation law is fulfilled for each fluid phase at both porous matrix and fracture domain; moreover, the mass exchange between the matrix and fracture is guaranteed through the integral formulation of mass conservation law. Applying the X-FEM technique, the explicit representation of macro-cracks is modeled by enriching the standard finite element approximation space with an enrichment function. Finally, several numerical examples of CO₂ injection into a brine aquifer below a naturally fractured cap-rock are modeled in order to investigate the effects of cracks’ aperture and orientation as well as the temperature of aquifer and the depth of injection on the leakage pattern of the carbon dioxide through the cap-rock.

     

Sizing problem for a cross flow heat exchanger with wing-type vortex generator

In the present study, sizing of a single pass cross flow heat exchanger with unmixed fluid streams has been investigated. The heat exchanger is a cross flow heat exchanger. It has overall dimensions of 20 × 20 × 20 cm. Two the most common heat exchanger design problems are the rating and sizing problem. Sizing problems deal with designing an exchanger and determining its physical size to meet the specified heat duty, pressure drops and other considerations. It means the determination of the exchanger construction type, flow arrangement, heat transfer surface geometries and materials, and the physical sizes of an exchanger to meet specified heat transfer and pressure drop. In this study, the physical size (length, width, height, mass flow rates of both fluids and surface areas on each side of the exchanger) are determined. Inputs to the sizing problem are surface geometries, fluid mass flow rates, inlet and outlet fluid temperatures and pressure drop on each side. Dimensions of L _a , L _b , and L _c for the selected surfaces were investigated such that the design meets the heat duty and pressure drops on both sides exactly.     

An experimental study on buoyancy-driven convective mass transfer from spheres embedded in saturated porous media

Buoyancy-driven convective mass transfer coefficients from copper spheres embedded saturated porous media have been experimentally obtained. Limiting diffusion current technique based on cathodic reduction of cupric ions is used. The data correspond to high Rayleigh number and, expectedly, exhibit non-Darcian effects. It is found that mass transfer from the spheres embedded in saturated porous media can be predicted by Sh^* _D =5.609Ra^*0.241 _D Da^−0.214, where Ra^* _D and Da are modified Rayleigh number and Darcy's number respectively. Received on 2 March 1999     

Effects of anisotropy in permeability on the two-phase flow and heat transfer in a porous cavity

This paper reports on the results of a numerical study of convection flow and heat transfer in a rectangular porous cavity filled with a phase change material under steady state conditions. The two vertical walls of the cavity are subject respectively to temperatures below and above the melting point of the PCM while adiabatic conditions are imposed on the horizontal walls. The porous medium is characterized by an anisotropic permeability tensor with the principal axes arbitrarily oriented with respect to the gravity vector. The problem is governed by the aspect ratioA, the Rayleigh numberRa, the anisotropy ratioR and the orientation angle θ of the permeability tensor. Attention is focused on these two latter parameters in order to investigate the effects of the anisotropic permeability on the fluid flow and heat transfer of the liquid/solid phase change process. The method of solution is based on the control volume approach in conjunction with the Landau-transformation to map the irregular flow domain into a rectangular one. The results are obtained for the flow field, temperature distribution, interface position and heat transfer rate forA=2.5,Ra=40, 0≤θ≤π, 0.25≤R≤4. It was found that the equilibrium state of the solid/liquid phase change process may be strongly influenced by the anisotropy ratioR as well as by the orientation angle θ of the permeability tensor. First, for a given set of parametersA,Ra andR, there exists an optimum orientation θ_max for which the flow strength, the liquid volume and the heat transfer rate are maximum. There also exists an orientation θ_min=θ_max+π/2 for which these quantities are minimum. Second, when an anisotropic medium is oriented along the optimum direction θ_max, an increase of the permeability component along that direction will increase the flow and heat transfer rate in a same order while an increase of the other permeability component only has a negligible effect. For the parameter ranges considered in the present study, it was found that the optimum direction is lying between the gravity vector and the dominant flow direction.     

Contribution to the Modelling of Coupled Heat and Mass Transfers on 3D Real Structure of Heterogeneous Building Materials: Application to Hemp Concrete

Hydraulic flow, electrical flow and the passage of elastic waves through porous media are all linked by electrokinetic processes. In its simplest form, the passage of elastic waves through the porous medium causes fluid to flow through that medium and that flow gives rise to an electrical streaming potential and electrical counter-current. These processes are frequency-dependent and governed by coupling coefficients which are themselves frequency-dependent. The link between fluid pressure and fluid flow is described by dynamic permeability, which is characterised by the hydraulic coupling coefficient (C_hp). The link between fluid pressure and electrical streaming potential is characterised by the streaming potential coefficient (C_sp). While the steady-state values of such coefficients are well studied and understood, their frequency dependence is not. Previous work has been confined to unconsolidated and disaggregated materials such as sands, gravels and soils. In this work, we present an apparatus for measuring the hydraulic and streaming potential coefficients of high porosity, high permeability consolidated porous media as a function of frequency. The apparatus operates in the range 1 Hz to 2 kHz with a sample of 10 mm diameter and 5–30 mm in length. The full design and validation of the apparatus are described together with the experimental protocol it uses. Initial data are presented for three samples of Boise sandstone, which present as dispersive media with the critical transition frequency of 918.3?±?99.4 Hz. The in-phase and in-quadrature components of the measured hydraulic and streaming potential coefficients have been compared to the Debye-type dispersion model as well as theoretical models based on bundles of capillary tubes and porous media. Initial results indicate that the dynamic permeability data present an extremely good fit to the capillary bundle and Debye-type dispersion models, while the streaming potential coefficient presents an extremely good fit to all of the models up to the critical transition frequency, but diverges at higher frequencies. The streaming potential coefficient data are best fitted by the Pride model and its Walker and Glover simplification. Characteristic pore size values calculated from the measured critical transition frequency fell within 1.73% of independent measures of this parameter, while the values calculated directly from the Packard model showed an underestimation by about 12%.

     

Radiative and Thermal Dispersion Effects on Non-Darcy Natural Convection with Lateral Mass Flux for Non-Newtonian Fluid from a Vertical Flat Plate in a Saturated Porous Medium

The problem of natural convective heat transfer for a non-Newtonian fluid from an impermeable vertical plate embedded in a fluid-saturated porous medium has been analyzed. Non-Darcian, radiative and thermal dispersion effects have been considered in the present analysis. The governing boundary layer equations and boundary conditions are cast into a dimensionless form and simplified by using a similarity transformation. The resulting system of equations is solved by using a double shooting Runge–Kutta method. The effect of viscosity index n, the conduction–radiation parameter R, the non-Darcy parameter Gr*, the thermal dispersion parameter Ds and the suction/injection parameter f_w on the fluid velocities, temperatures and the local Nusselt number are discussed.     

Effects of chemical reaction on mixed convection flow of a polar fluid through a porous medium in the presence of internal heat generation

This paper reports a detailed numerical investigation on mixed convection flow of a polar fluid through a porous medium due to the combined effects of thermal and mass diffusion. The energy equation accounts for heat generation or absorption, while the nth order homogeneous chemical reaction between the fluid and the diffusing species is included in the mass diffusion equation. The governing equations of the linear momentum, angular momentum, energy and concentration are obtained in a non-similar form by introducing a suitable group of transformations. The final set of non-similar coupled non-linear partial differential equations is solved using an implicit finite-difference scheme in combination with quasi-linearization technique. The effects of various parameters on the velocity, angular velocity, temperature and concentration fields are investigated. Numerical results for the skin friction coefficient, wall stress of angular velocity, Nusselt number and Sherwood number are also presented.     

Flow and Heat Transfer of Viscous Fluids Saturated in Porous Media over a Permeable Non-isothermal Stretching Sheet

The present paper deals with the flow and heat transfer of a viscous fluid saturated in a porous medium past a permeable and non-isothermal stretching sheet with internal heat generation or absorption and radiation. Closed-form solutions to steady, two dimensional momentum equations with neglecting quadratic inertia terms and heat transfer equation are found using a similarity transformation. Asymptotic expressions of the temperature functions are also presented valid for both very large and very small modified Prandtl numbers. Attention is focused on the effects of porous parameter K, suction parameter R, radiation parameter Nr, viscosity ratio Λ, internal heat parameter α and Prandtl number P to the characteristics of flow and heat transfer.