non-Darcy Model Research Articles

Difference Schemes for Three Kinds of Nonlinear Flow Model in Low Permeability Porous Media

This paper discusses three kinds of typical one-dimensional nonlinear equations coming from low permeability reservoir seepage models with different boundary conditions. The several finite difference methods including forward difference method and second central order difference quotient method are used for the respective discrete process of three models. With these difference methods, the discrete schemes of models are obtained. Then the corresponding nonlinear discrete equations are deduced. While dealing with the boundary condition, the mid-rectangle formula is used. Finally, integrated discrete equations of three nonlinear equations are formed. The results should be meaningful for the numerical simulation of non-Darcy flow model of the low-permeability oil wells.

DRBEM solution of natural convective heat transfer with a non-Darcy model in a porous medium

This study presents the dual reciprocity boundary element (DRBEM) solution of Brinkman–Forchheimer-extended Darcy model in a porous medium containing an incompressible, viscous fluid. The governing dimensionless equations are solved in terms of stream function, vorticity and temperature. The problem geometry is a unit square cavity with either partially heated top and bottom walls or hot steps at the middle of these walls. DRBEM provides one to obtain the expected behavior of the flow in considerably small computational cost due to the discretization of only the boundary, and to compute the space derivatives in convective terms as well as unknown vorticity boundary conditions using coordinate matrix constructed by radial basis functions. The Backward-Euler time integration scheme is utilized for the time derivatives. The decrease in Darcy number suppresses heat transfer while heat transfer increases for larger values of porosity, and the natural convection is pronounced with the increase in Rayleigh number.

A thermal non-equilibrium approach on double-diffusive natural convection in a square porous-medium cavity

In this manuscript the influence of local thermal non-equilibrium state on double-diffusive natural convection in a square cavity filled with fluid-saturated porous medium has been addressed numerically. The two dimensional steady state flow is induced due to maintenance of constant temperature and concentration on the vertical walls and insulation of both horizontal walls of the cavity. Non-Darcy (Darcy–Brinkman–Forchheimer) model has been taken and the complete governing equations are solved by standard SIMPLER algorithm. A comparative study of the effect of the presence of Brinkman term in the momentum equation showed that results under the Darcy model are very close to those for the non-Darcy Brinkman model for relatively low permeable medium (e.g., in this study Da=10-4). From our experiments it has been found that the impact of Lewis number (Le) on the average heat transfer rate of fluid (Nuf) and solid (Nus) as well as on the thermal distribution of fluid and solid is not straightforward. It depends on the buoyancy ratio (N) and the value of inter-phase heat transfer coefficient (H). However, Le increases the average mass transfer rate (Sh). Also, for each Le there exist a point in the domain of N where Nuf is minimum. Similar points also exist for Nus and Sh. In general, these points are different and depend on the LTNE state parameters, except at Le=1. For any relatively large value of H, when almost thermal equilibrium state is achieved, the point at which Nuf and Nus are minimum is same due to similar thermal distribution of fluid and solid. Also, it has been found that, for the buoyancy aided flow (N>0), the increase in H up to a threshold value (H0) decreases Sh as well as Nuf but increases Nus. The H0 is found to be a decreasing function of the porosity scaled thermal conductivity ratio (γ) of fluid and solid phases. Overall, the impact of LTNE state on the heat transfer rate and thermal distribution is significant but it is negligible on the mass transfer rate and solute distribution.

An incompressible smoothed particle hydrodynamics method for natural/mixed convection in a non-Darcy anisotropic porous medium

In this paper, the non-Darcy model for natural/mixed convection and heat transfer in a cavity saturated with anisotropic porous media has been investigated using an incompressible smoothed particle hydrodynamics (ISPH) method. In the ISPH algorithm, a semi-implicit velocity correction procedure has been used and the pressure is obtained by solving pressure Poisson equation. The unsteady natural/mixed convection in non-Darcy porous cavities is examined by our extended ISPH method and Finite Volume Method. The results are presented with flow configurations, isotherms and average Nusselt numbers for different Darcy numbers from 10−4 to 10−2, porosity values from 0.4 to 0.9, permeability ratio 0.1–10, inclination angle of permeability 0–90° and Reynolds/Rayleigh numbers. The flow pattern and rate of heat transfer inside the cavity are affected by these parameters. The results demonstrate the effects of the parameters such as Darcy number, porosity, permeability ratio and inclination angle in both of the heat transfer rate and the flow regime. The results from this investigation are well validated and have favorable comparisons with previously published results.

Unsteady Conjugate Natural Convection in a Vertical Cylinder Containing a Horizontal Porous Layer: Darcy Model and Brinkman-Extended Darcy Model

Transient natural convection in a vertical cylinder partially filled with a porous media with heat-conducting solid walls of finite thickness in conditions of convective heat exchange with an environment has been studied numerically. The Darcy and Brinkman-extended Darcy models with Boussinesq approximation have been used to solve the flow and heat transfer in the porous region. The Oberbeck–Boussinesq equations have been used to describe the flow and heat transfer in the pure fluid region. The Beavers–Joseph empirical boundary condition is considered at the fluid–porous layer interface with the Darcy model. In the case of the Brinkman-extended Darcy model, the two regions are coupled by equating the velocity and stress components at the interface. The governing equations formulated in terms of the dimensionless stream function, vorticity, and temperature have been solved using the finite difference method. The main objective was to investigate the influence of the Darcy number \(10^{-5}\le \hbox {Da}\le 10^{-3}\), porous layer height ratio \(0\le d/L\le 1\), thermal conductivity ratio \(1\le k_{1,3}\le 20\), and dimensionless time \(0\le \tau \le 1000\) on the fluid flow and heat transfer on the basis of the Darcy and non-Darcy models. Comprehensive analysis of an effect of these key parameters on the Nusselt number at the bottom wall, average temperature in the cylindrical cavity, and maximum absolute value of the stream function has been conducted.

Non-Darcy displacement in linear composite and radial aquifer during CO<SUB align="right">2 sequestration

This paper presents Buckley-Leverett type analytical solutions for non-Darcy displacement of two immiscible fluids in linear and radial composite porous media. High velocity or non-Darcy flow commonly occurs in the vicinity of the wellbore because of smaller flowing cross-sectional areas. However, the effect of such non-Darcy flow has been traditionally ignored. To examine the physical behaviour of multiphase immiscible fluids in non-Darcy displacement, an extended Buckley-Leverett type of solution is discussed. There exists a Buckley-Leverett type solution for describing non-Darcy displacement in a linear homogeneous reservoir. This work extends the solution to flow in linear and radial composite flow systems. We present several new Buckley-Leverett type analytical solutions for non-Darcy flow in more complicated flow geometries of linear and radial composite reservoirs, based on non-Darcy flow models of Forchheimer and Barree-Conway. As application examples, we use the analytical solutions to verify numerical simulation results as well as to discuss non-Darcy displacement behaviour. This theory of non-Darcy flow displacement is applied to evaluate the flow behaviour near wellbore areas during CO2 sequestration. The results show how non-Darcy displacement during CO2 injection in linear and radial composite systems is controlled not only by relative permeability, but also by non-Darcy coefficients, characteristic length, injection rates, as well as discontinuities in the saturation profile across the interfaces between adjacent composite flow domains.

Study on the Unstable Seepage Flow Model and the Productivity Characteristics in Tight Gas Heterogeneity Reservoir

In this paper, considering on the real threshold pressure gradient (TPG), the mathematical model for low velocity non-Darcy unstable gas flow was established for the tight gas reservoir. Based on the low velocity non-Darcy unstable gas flow model, the gas flow model in heterogeneity formation was developed, and the analytical solution was obtained on the condition of constant production for inner boundary. Then we calculated the model numerically by Matlab programming. Its analyzed that the pressure distribution and the influencing factors such as production rate, producing time, reservoir thickness and the threshold pressure gradient. We compared the heterogeneity formation with the homogeneous formation by the heterogeneity low velocity non-Darcy unstable gas flow model, which shows great difference and illustrate the significance of research on heterogeneity.

Analysis of Convective Heat Flow Visualization within Porous Right Angled Triangular Enclosures with a Concave/Convex Hypotenuse

Analysis of natural convection in porous right angled triangular enclosures with a concave/convex hypotenuse has been carried out using the Bejan's heatlines approach. A generalized non-Darcy model without Forchheimer term is employed for fluid flow in a porous matrix and the governing equations are solved by the Galerkin finite element method. The cavity is subjected to a thermal boundary condition of an isothermal cold left wall, isothermal hot curved right wall, and adiabatic bottom wall. Due to intense closed loop heatlines, thermal mixing is higher in convex cases compared to the concave case for all parameters. Average heat transfer rate is found to be largest in the concave hypotenuse case.

Pore-to-core simulations of flow with large velocities using continuum models and imaging data

We consider computational modeling of flow with small and large velocities at porescale and at corescale, and we address various challenges in simulation, upscaling, and modeling. While our focus is on voxel-based data sets from real porous media imaging, our methodology is verified first on synthetic geometries, and we analyze various scaling and convergence properties. We show that the choice of a voxel-based grid and representative elementary volume size can lead up to 10–20 % difference in calculated conductivities. On the other hand, the conductivities decrease significantly with flow rates, starting in a regime usually associated with the onset of inertia effects. This is accompanied by deteriorating porescale solver performance, and we continue our experiments up until about 50 % reduction in conductivities, i.e., to Reynolds number just under 1. To account for this decrease, we propose a practical power-based fully anisotropic non-Darcy model at corescale for which we calculate the parameters by upscaling.

Analysis of Flow and Heat Transfer in a Vertical Rectangular Duct Using a Non-Darcy Model

Numerical investigation of steady natural convection flow through a fluid-saturated porous medium in a vertical rectangular duct is investigated. The Darcy-Forchheimer-Brinkman model is used to represent the fluid transport within the porous medium. One of the vertical walls of the duct is cooled to a constant temperature, while the other wall is heated to constant but different temperature. The other two sides of the duct are insulated. The finite difference method of second-order accuracy is used to solve the non-dimensional governing equations. The results are presented graphically to show the effects of the Darcy number, inertial parameter, Grashof number, Brinkman number, aspect ratio, and viscosity ratio. It is found that an increase in the Darcy number and inertial parameter reduces the flow intensity whereas an increase in the Grashof number, Brinkman number, aspect ratio, and viscosity ratio increases the flow intensity.

Effect of Buoyancy Ratio on Non-Darcy Mixed Convection in a Vertical Channel: A Thermal Non-equilibrium Approach

This article presents a numerical study of the doublediffusive mixed convection in a vertical channel filled with porous medium by using non-equilibrium model. The flow is assumed fully developed, uni-directional and steady state. The controlling parameters are thermal Rayleigh number (RaT ), Darcy number (Da), Forchheimer number (F), buoyancy ratio (N), inter phase heat transfer coefficient (H), and porosity scaled thermal conductivity ratio (γ). The Brinkman-extended non-Darcy model is considered. The governing equations are solved by spectral collocation method. The main emphasize is given on flow profiles as well as heat and solute transfer rates, when two diffusive components in terms of buoyancy ratio are in favor (against) of each other and solid matrix and fluid are thermally non-equilibrium. The results show that, for aiding flow (RaT = 1000), the heat transfer rate of fluid (Nuf ) increases upto a certain value of H, beyond that decreases smoothly and converges to a constant, whereas in case of opposing flow (RaT = -1000), the result is same for N = 0 and 1. The variation of Nuf in (N, Nuf )-plane shows sinusoidal pattern for RaT = -1000. For both cases (aiding and opposing) the flow destabilize on increasing N by inviting point of inflection or flow separation on the velocity profile. Overall, the buoyancy force have significant impact on the non-Darcy mixed convection under LTNE conditions. Keywords—buoyancy ratio, mixed convection, non-Darcy model, thermal non-equilibrium

Water seepage flow in concrete

The waterproofing durability of concrete is an attentive focus in underground engineering and hydraulic structures. Water seepage law in non-cracked concrete, concrete under tension, straight smooth surface cracks and real cracks in concrete were mainly studied in this paper. And the waterproofing capacity of concrete was analyzed based on the research results. The experimental results indicated that Non-Darcy model was followed in non-cracked concrete weather it bears tensile stress or not. There existed a balance depth in the process of the water permeation. Tensile stress could change the starting pressure gradient (λ) of concrete. With the increase of tension, the λ decreased. When the concrete was cracked, water infiltrated into the concrete through cracks, the linear Darcy Law was followed in this case. To characterize the water seepage law in the real concrete cracks, two parameters τ (tortuosity of cracks) and m (toughness of cracks) were introduced into the Navier–Stokes Equation. And the result indicated the modified Navier–Stokes Equation could describe the seepage law in real concrete cracks well and the value of m in the real concrete cracks was less than 1.15 according to the experimental results.

Asymptotic analysis of a Hiemenz flow in a low-porosity medium with phase change

AbstractIn the present work, the features of liquid evaporation inside a low-porosity medium subjected to an impinging stream of hot gas is investigated analytically. The flow is analysed for a non-Darcy model, in which viscous and convective terms are considered in the Darcy pressure equation. A low-volatility liquid is considered, so that a low-vaporization regime is established. The rates of heat transfer between gas and solid and between liquid and solid are assumed to be high. Owing to differences between phase properties, in the system under study, different physical processes occur at different length scales. Using asymptotic expansions, expressions for the three phases that occur in this problem are obtained, in each of their length scales. The results predict that high injection temperatures are needed for phase change to occur, as a result of the low volatility of the liquid. Likewise, the enhancement of the vaporization rate due to heat conduction in the porous medium is quantified. The Hiemenz flow pressure term is modified to incorporate the effect of the porous medium, which is necessary for a solution to be found.

Influence of periodicity of sinusoidal bottom boundary condition on natural convection in porous enclosure

In this paper, natural convective flow within a rectangular enclosure has been investigated numerically. All the walls of the enclosure are adiabatic except the bottom wall, which is partially heated and cooled by sinusoidal temperature profile. Both situations: medium is hydro-dynamically isotropic and anisotropic are considered. The governing equations are written under assumption of Brinkman-extended non-Darcy model, including material derivative, and then solved by numerically using spectral element method (SEM). Main emphasize is given on effect of periodicity parameter (N) on local heat transfer rate (Nux) as well as flow mechanism in the enclosure. The result shows that, as the periodicity is decreased on increasing N, the absolute value of Nux at the bottom left corner point increases in both situations. For odd values of N, the local heat transfer profile is symmetric about the line X=0.5. The entire flow is governed by two type convective cells: (i) rotating clockwise (ii) rotating anticlockwise. Furthermore, increasing of N increases the multiple cellular structures in the form of N+1. The relative impact of media permeability on Nux at (0,0) shows that enhancement of media permeability along Y-direction is much more effective than the same along X-direction. In contrast to isotropic porous media, the effect of the permeability and permeability orientation angle on the flow is complex and often non intuitive. In particular the present analysis shows that, different periodicity of temperature boundary condition has significant effect on the flow mechanism and consequently on the heat transfer rate for both situations.

Pressure Behavior of a Coupling Model with Variable Permeability Effect

It is indicated that the fluid flow does not abide by the Darcy law in certain low permeability reservoirs. The results of pressure date analysis in these low permeability reservoirs would be inaccurate when generalized using Darcy flow models. To improve analysis results, based on the concept of variable permeability effect, the dual non-Darcy coupling flow model is established in this paper. The iterative algorithm is adopted to solve the differential set of equations. This work also describes the pressure behaviors by pressure response curves.

Pressure Behavior of a Coupling Model with Variable Permeability Effect

It is indicated that the fluid flow does not abide by the Darcy law in certain low permeability reservoirs. The results of pressure date analysis in these low permeability reservoirs would be inaccurate when generalized using Darcy flow models. To improve analysis results, based on the concept of variable permeability effect, the dual non-Darcy coupling flow model is established in this paper. The iterative algorithm is adopted to solve the differential set of equations. This work also describes the pressure behaviors by pressure response curves.

Influence of Non-Uniform Sinusoidal Periodic Bottom Boundary Condition on Natural Convection in an Isotropic Porous Enclosure

A comprehensive numerical investigation on the natural convection in an isotropic porous enclosure is presented. All the walls of the enclosure are adiabatic except the bottom wall which is partially heated and cooled by sinusoidal temperature profile. The governing equations were written under assumption of Brinkman-extended non-Darcy model, including material derivative, and then solved by numerically using spectral element method (SEM). The heat transfer and fluid flow mechanisms in isotropic case are governed by periodicity parameter (N) Rayleigh Number (Ra), Darcy number (Da), aspect ratio (A), Prandtl number (Pr) and media permeability (K). The main emphasize is given on effect of N on local heat transfer as well as mechanism of heat transfer and fluid flow in enclosure. The results shows that, as the periodicity is decreased on increasing N the absolute value of Nux at the bottom left corner point increases. For odd values of N, the local heat transfer profile is symmetric about the line x=0.5, which is consequence of symmetric boundary condition at the bottom wall of the enclosure. The entire flow is governed by two type convective cells: (i) rotating clockwise (ii) rotating anticlockwise. Furthermore for even values of N cells rotating anticlockwise are dominated and covered the entire domain. In particular the present analysis shows that, different periodicity of temperature boundary condition has the significant effect on the flow mechanism and consequently on the heat transfer rate.

Effects of Non-Darcian and Inlet Conditions on the Forced Convection Along a Vertical Plate With Film Evaporation

The present study analyzes theoretically the non-Darcian effects and inlet conditions of forced convection flow with liquid film evaporation in a porous medium. The physical scheme includes a liquid–air streams combined system; the liquid film falls down along the plate and is exposed to a cocurrent forced moist air stream. The axial momentum, energy, and concentration equations for the air and water flows are developed based on the steady two-dimensional (2-D) laminar boundary layer model. The non-Darcian convective, boundary, and inertia effects are considered to describe the momentum characteristics of a porous medium. The paper clearly describes the temperature and mass concentration variations at the liquid–air interface and provides the heat and mass transfer distributions along the heated plate. Then, the paper further evaluates the non-Darcian effects and inlet conditions on the heat transfer and evaporating rate of liquid film evaporation. The numerical results show that latent heat transfer plays the dominant heat transfer role. Carrying out a parametric analysis indicates that higher air Reynolds number, higher wetted wall temperature, and lower moist air relative humidity will produce a better evaporating rate and heat transfer rate. In addition, a non-Darcy model should be adopted in the present study. The maximum error for predictions of heat and mass transfer performance will be 21% when the Darcy model is used.

Heatline Analysis for Natural Convection within Porous Rhombic Cavities with Isothermal/Nonisothermal Hot Bottom Wall

Analysis has been carried out for energy distribution and thermal mixing in steady laminar natural convective flow through the porous rhombic cavities with inclination angle φ for various applications such as geothermal, grain storage, electronic cooling, etc. A generalized non-Darcy model without the Forchheimer inertia term is used to predict the flow in porous media. The effect of the Darcy number (Da) and the role of φ on the energy distribution and thermal mixing within porous rhombic cavities with isothermal (case 1) and nonisothermal (case 2) hot bottom walls are illustrated via “heatlines”. Heat transfer is found to be primarily conduction dominant at Da = 10–5 even at a higher Rayleigh number (Ra = 106). The onset of convection occurs at Da = 10–4, and the distorted heatlines from the hot bottom wall take a longer path to reach the cold side walls of the cavity. Larger heat transfer and thermal mixing occurs for Da = 10–3 at Ra = 106 irrespective of φ and Pr. Multiple flow/convective circulations...

Simulation of Non-Darcy Porous Media Flow According to the Barree and Conway Model

Non-Darcy porous media flow has been traditionally handled using the Forchheimer equation. However, recent experimental studies have shown that the Forchheimer model is unable to fit laboratory results at high flow rates. On the other hand, the non-Darcy flow model, proposed by Barree and Conway, is capable of describing the entire range of relationships between flow rate and potential gradient from low- to high-flow rates through proppant packs. In this paper, we present a numerical model by incorporating the Barree and Conway model into a general-purpose reservoir simulator for modeling single-phase and multiphase non-Darcy flow in porous and fractured media. The numerical formulation is based on the TOUGH2 methodology, i.e., spatial integral-finite-difference discretization, leading to an unstructured grid, followed by time discretization carried out with a backward, first-order, finite-difference method. The final discrete nonlinear equations are handled fully implicitly by Newton iteration. In the numerical approach, flow through fractured rock is handled using a general multi-continuum approach, applicable to both continuum and discrete fracture conceptual models. In an effort for model validation, we use analytical solutions to verify our numerical model results for both single-phase and multiphase non-Darcy flow. In addition, the numerical model is applied for well testing analysis of transient non-Darcy flow toward a well.