Flow In Anisotropic Porous Media Research Articles

Nano‐bioconvective anisotropic slip flow in anisotropic porous medium with Coriolis force effects

AbstractThe present paper examines the impact of the Coriolis force and anisotropic Darcian porous medium on the nonsteady convective flow of bio‐nanofluid along a three‐dimensional stretchable surface. The problem has applications in extrusion processes and the geographical movement of tectonic plates under a moveable ocean. Emphasis is placed on the combined effects of anisotropic porous medium, anisotropic momentum slip and temperature, and microorganism slips at the wall. The pertaining equations are condensed to a set of similarity equations before being simulated using the Keller‐box method, an efficient numerical technique. The study reveals that the friction factor along the y‐axis decreases as the rotation parameter increases. Moreover, the heat and nanoparticle volume fraction movement gradient decrease along the y‐axis when the Darcy number and stretching rate increase. However, these rates increase with a higher velocity slip. In addition, the density of microorganisms in the bio‐nanofluid decreases when there are higher levels of unsteadiness, rotation, and Darcy numbers, as well as with increased bioconvection parameters.

Physics of generalized couette flow of immiscible fluids in anisotropic porous medium

This study presents an investigation on the generalized Couette flow of two immiscible Newtonian fluids in an anisotropic porous medium. The flow occurs between two parallel plates, driven by the combined effects of an axial pressure gradient and the movement of the upper plate. The region between the plates is filled with an anisotropic porous medium characterized by directional permeability. An exact solution is derived for the proposed problem by considering appropriate boundary and interface conditions. The slip condition is employed at the lower plate of the channel which has a significant role in the flow mechanics of fluids flowing through porous medium. The effects of directional permeability on the flow of immiscible fluids are explored, and specific cases are presented. This study reveals a significant influence of directional permeability on the flow velocity of both fluids. Furthermore, quantitative conclusions are drawn regarding the shear stress at the lower and upper wall of the channel, with implications for the anisotropic nature of blood arteries. The findings contribute to a deeper understanding of fluid flow in anisotropic porous media and have potential applications in various engineering domains.

Generalized finite difference method-based numerical modeling of oil–water two-phase flow in anisotropic porous media

This paper applies generalized finite difference method (GFDM) to a compressible two-phase flow in anisotropic porous media with the aim of further extending the wider application of this class of meshless methods. We develop an implicit Euler scheme in time and a GFDM discretization in space based on two treatments of the anisotropic permeability tensor in continuous function expression and discrete distribution. The effectiveness and generality of GFDM for two-phase flow problems in anisotropic porous media are verified by three examples with rectangular, irregular, and complex boundaries. Also, the computational performance of the method is verified according to the error calculation with L2 absolute error functions in different node collocation schemes. In addition, the sensitivity analysis of the radius of the influence domain to the transient pressure equation (parabolic equation) and the saturation equation (hyperbolic equation) is considered. It generally holds that the larger the radius of the influence domain, the lower the calculation accuracy in the case of Cartesian collocation. This may be a preliminary rule for the radius choice of the influence domain for GFDM. In sum, this work provides an efficient and accurate meshless solver to handle two-phase flow problems in anisotropic porous media under the GFDM framework, which reveals the great application potential of GFDM in reservoir numerical simulation.

The dimensional character of permeability: Dimensionless groups that govern Darcy's flow in anisotropic porous media

AbstractThe dimensional character of permeability in anisotropic porous media, that is, its dimension or dimensional equation, is an information that allows setting the dimensionless groups that govern the solution of the flow equation in terms of hydraulic potential patterns. However, employing the dimensional basis {L, M, T} (length, mass, time), the dimensionless groups containing the anisotropic permeability do not behave as independent monomials that rule the solutions. In this work, the contributions appearing in the literature on the dimensional character of permeability are discussed and a new approach based on discriminated and general dimensional analysis is presented. This approach leads to the emergence of a new and accurate dimensionless group, , a ratio of permeabilities corrected by the squared value of an aspect factor, being and two arbitrary lengths of the domain in the directions that are indicated in their subscripts. Specific values of this lengths, which we name ‘hidden characteristic lengths’, are also discussed in this article. To check the validity of this dimensionless group, numerical simulations of two illustrative 2‐D seepage scenarios have been solved.

A new and consistent well model for one-phase flow in anisotropic porous media using a distributed source model

A new well model for one-phase flow in anisotropic porous media is introduced, where the mass exchange between well and a porous medium is modeled by spatially distributed source terms over a small neighborhood region. To this end, we first present a compact derivation of the exact analytical solution for an arbitrarily oriented, infinite well cylinder in an infinite porous medium with anisotropic permeability tensor in R3, for constant well pressure and a given injection rate, using a conformal map. The analytical solution motivates the choice of a kernel function to distribute the sources. The presented model is independent from the discretization method and the choice of computational grids. It can be applied to porous media with full anisotropic permeability tensors. In numerical experiments, the new well model is shown to be consistent and robust with respect to rotation of the well axis, rotation of the permeability tensor, and different anisotropy ratios. Finally, a comparison with a Peaceman-type well model suggests that the new scheme leads to an increased accuracy for injection (and production) rates for arbitrarily-oriented pressure-controlled wells.

WITHDRAWN: A new and consistent well model for one-phase flow in anisotropic porous media using a distributed source model

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Positivity-preserving finite volume scheme for compressible two-phase flows in anisotropic porous media: The densities are depending on the physical pressures

We are concerned with the approximation of solutions to a compressible two-phase flow model in porous media thanks to an enhanced control volume finite element discretization. The originality of the methodology consists in treating the case where the densities are depending on their own pressures without any major restriction neither on the permeability tensor nor on the mesh. Contrary to the ideas of [23], the point of the current scheme relies on a phase-by-phase “sub”-unpwinding approach so that we can recover the coercivity-like property. It allows on a second place for the preservation of the physical bounds on the discrete saturation. The convergence of the numerical scheme is therefore performed using classical compactness arguments. Numerical experiments are presented to exhibit the efficiency and illustrate the qualitative behavior of the implemented method.

Modeling flow in anisotropic porous medium with full permeability tensor

The flow in anisotropic porous medium is significant for the modeling of subsurface fluids transportation. The subsurface porous medium is usually both heterogeneous and anisotropic, caused by the compaction and sedimentations effects on the formation. Full permeability tensor is therefore needed in modeling flow in anisotropic medium. In this research, two widely used finite volume schemes, Two-Point Flux Approximation (TPFA) and Multi-Point Flux Approximation (MPFA), are applied to solve the flow model with a full permeability tensor. The results verified that ignoring anisotropy of the porous medium results in overestimation of the total flux. The TPFA methods have high computational efficiency, but failed to represent the anisotropy. The MPFA scheme takes more CPU time than TPFA for same grid block resolution, but incorporates the anisotropy using a full tensor. The comparison between the results from two methods indicates that ignoring anisotropy results in significant errors in determined flux.

Positive control volume finite element scheme for a degenerate compressible two-phase flow in anisotropic porous media

In this paper, we are concerned with the convergence analysis of a positive control volume finite element scheme (CVFE) for a degenerate compressible two-phase flow model in anisotropic porous media. For this, we consider the global pressure saturation formulation. We next use an implicit Euler scheme in time and a CVFE discretization in space. This approach rests on a particular choice of the mean value of the gas density on the interfaces, a centered scheme of the total mobility, and the upwind approximation of fractional fluxes according to the total velocity. Thus, the maximum principle is fulfilled without any constraint on the stiffness coefficients. Moreover, uniform estimates on the discrete gradient of the global pressure and the dissipative term are derived. As the mesh size is sent to zero, we establish that the sequence of approximate solutions converges to a weak solution of the continuous problem. Numerical tests are presented in two dimensions to exhibit the behavior of the gas pressure and the water saturation through the medium.

An implicit algorithm of solving Navier–Stokes equations to simulate flows in anisotropic porous media

A coupled computational algorithm for modified Navier–Stokes equations to simulate flows in anisotropic porous media is proposed and described. The difference from the classic SIMPLE algorithm is in the completely implicit relationship between velocity and pressure owing to the implicit terms of the pressure and mass flow gradients in the continuity equation and momentum equation. One of the attractive features of this algorithm is the possibility of completely implicit discretization of off-diagonal components of the porous medium resistance tensor in the right-hand side of the momentum equation. Implicit discretization allows reducing the number of linear iterations as compared to the SIMPLE algorithm with explicit discretization of off-diagonal components. The specific features of discretization of modified equations including the discretization of boundary conditions and components of the porous medium resistance tensor are considered. The proposed algorithm is verified in comparison with the SIMPLE algorithm on a series of benchmark problems, such as the problem of a flow through a porous insert, a flow in a divided channel, and a flow through a cylindrical porous filter. The total problem runtimes and the number of iterations required for complete convergence are given to compare the two algorithms.

Multiphase flow simulation with gravity effect in anisotropic porous media using multipoint flux approximation

Numerical investigations of two-phase flows in anisotropic porous media have been conducted. In the flow model, the permeability has been considered as a full tensor and is implemented in the numerical scheme using the multipoint flux approximation within the framework of finite difference method. In addition, the experimenting pressure field approach is used to obtain the solution of the pressure field, which makes the matrix of coefficient of the global system easily constructed. A number of numerical experiments on the flow of two-phase system in two-dimensional porous medium domain are presented. In this work, the gravity is included in the model to capture the possible buoyancy-driven effects due to density differences between the two phases. Different anisotropy scenarios have been considered. From the numerical results, interesting patterns of the flow, pressure, and saturation fields emerge, which are significantly influenced by the anisotropy of the absolute permeability field. It is found that the two-phase system moves along the principal direction of anisotropy. Furthermore, the effects of anisotropy orientation on the flow rates and the cross flow index are also discussed in the paper.

Definition of the capillary number for two-phase filtration flows in anisotropic porous media

Definition of the capillary number for two-phase filtration flows in anisotropic porous media

Solving global problem by considering multitude of local problems: Application to fluid flow in anisotropic porous media using the multipoint flux approximation

In this work we apply the experimenting pressure field approach to the numerical solution of the single phase flow problem in anisotropic porous media using the multipoint flux approximation. We apply this method to the problem of flow in saturated anisotropic porous media. In anisotropic media the component flux representation requires, generally multiple pressure values in neighboring cells (e.g., six pressure values of the neighboring cells is required in two-dimensional rectangular meshes). This apparently results in the need for a nine points stencil for the discretized pressure equation (27 points stencil in three-dimensional rectangular mesh). The coefficients associated with the discretized pressure equation are complex and require longer expressions which make their implementation prone to errors. In the experimenting pressure field technique, the matrix of coefficients is generated automatically within the solver. A set of predefined pressure fields is operated on the domain through which the velocity field is obtained. Apparently such velocity fields do not satisfy the mass conservation equations entailed by the source/sink term and boundary conditions from which the residual is calculated. In this method the experimenting pressure fields are designed such that the residual reduces to the coefficients of the pressure equation matrix.

Coupling Two-Phase Fluid Flow with Two-Phase Darcy Flow in Anisotropic Porous Media

This paper reports a numerical study of coupling two-phase fluid flow in a free fluid region with two-phase Darcy flow in a homogeneous and anisotropic porous medium region. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations in the free fluid region and the two-phase Darcy law in the anisotropic porous medium region. A Robin-Robin domain decomposition method is used for the coupled Navier-Stokes and Darcy system with the generalized Beavers-Joseph-Saffman condition on the interface between the free flow and the porous media regions. Obtained results have shown the anisotropic properties effect on the velocity and pressure of the two-phase flow.

Numerical study on the permeability in a tensorial form for laminar flow in anisotropic porous media

Pore-scale flow simulations were conducted to investigate the permeability tensor of anisotropic porous media constructed using the Voronoi tessellation method. This construction method enabled the introduction of anisotropy to the media at the particle level in a random and yet controllable way. Simulations were carried out for media with different degrees of anisotropy through varying the mean aspect ratio of grain particles. The simulation results were then analyzed using the Kozeny-Carman (KC) model. The KC model describes the permeability of the anisotropic media in a tensor form with the anisotropy represented by different tortuosities along the three principal directions. The tortuosity tensor was found to be a complex function of the particle morphology, which is yet to be fully determined. However, the results presented have established the starting point for further theoretical development to formulate such a function and to build closed-form analytical permeability models for anisotropic porous media based on first principles.

Weakly singular integral equations for Darcy's flow in anisotropic porous media

The weakly singular, weak-form fluid pressure and fluid flux integral equations are developed for steady-state Darcy's flow in a porous media which possesses generally anisotropic permeability. A systematic technique is established first to regularize the conventional fluid pressure and fluid flux integral equations in which the former contains a singular kernel (of order 1/ r 2) and the latter contains a strongly singular kernel (of order 1/ r 3). The key step in the regularization procedure is to construct decompositions for the fluid velocity fundamental solution and the strongly singular kernel in a form well-suited for performing an integration by parts via Stokes' theorem. Weakly singular kernels appearing in these decompositions are obtained in an explicit form by the Radon transform method. The final integral equations possess following important features: (1) they contain only weakly singular kernels of order 1/ r; (2) their validity requires only the continuity of pressure boundary data; and (3) they are applicable for modeling steady-state flow in porous media possessing generally anisotropic permeability and containing surfaces of discontinuity. A proper use of this set of weak-from integral equations forms a symmetric formulation for a weakly singular, symmetric Galerkin boundary element method (SGBEM). The numerical implementation is then carried out and certain examples are solved to demonstrate the accuracy of the method.

Non-Darcy-Flow Studies in Anisotropie Porous Media

Summary The velocity-pressure drop relationship in the near-wellbore region of high capacity gas and condensate reservoirs often deviates from Darcy's law. Furthermore, many naturally occurring porous media are anisotropic and often layered. This study is directed at combining these facts in developing a mechanistic understanding of non-Darcy gas flow in anisotropic porous media. Non-Darcy flow coefficients, permeabilities, and electrical tortuosities are measured in cores with layers both in the parallel and perpendicular directions to flow and in nonlayered, anisotropic cores. The presence of a connate water saturation decreases permeability in all cores but increases the non-Darcy coefficient significantly only in the perpendicular core. Both a macroscopic and a pore-scale network (microscopic) model of anisotropy are proposed. The product of permeability and non-Darcy coefficient is shown to be less anisotropic than either the permeability or the non-Darcy coefficient tensors in both experiments and simulations. In the microscopic model, the non-Darcy term has been found to be tensorial and proportional to the square of the superficial velocity at the macroscopic scale. As the pore-scale anisotropic parameter increases, the non-Darcy coefficient and tortuosity increase, but permeability decreases. Models show order of magnitude agreement with experimental data.

Modeling the effects of a magnetic field or rotation on flow in a porous medium: momentum equation and anisotropic permeability analogy

It is pointed out that, in recent publications, several authors have incorrectly omitted a porosity factor, from a term modeling the effect of a magnetic field or rotation, in the momentum equation modeling flow in a porous medium. The error is linked with the way in which the pressure is incorporated in the standard differential form of Darcy’s law, and this is discussed in detail. A new form for this equation is proposed. Also, the analogy between: (i) Darcy flow in an isotropic porous medium with a magnetic field or rotation effect present; and (ii) flow in a medium with anisotropic permeability, is discussed.

Determining equations of two-phase flows through anisotropic porous media

A new interpretation of the concept of relative phase permeability is given. Relative phase permeabilities are represented in the form of fourth-rank tensors. It is shown that in the case of anisotropic porous media functions depending not only on the saturation but also on the anisotropy parameters represented in the form of ratios of the principal values of the absolute permeability coefficient tensor correspond to the classical representation of the relative phase permeabilities. For a two-phase flow in anisotropic porous media with orthotropic and transversely-isotropic symmetry a generalized two-term Darcy’s law is analyzed.

Experimental and computational study of channelized water waves generated by a porous body

We report about experiments on large amplitude shallow water waves that were generated by the sudden forward motion of a porous wedge-type piston. The experiments are compared with those done previously [7] when the motion of a body with impervious boundary was examined. Close agreement is obtained between the two sets of experiments if the porous body is sealed at its rear side, but differences are substantial without the seal. The theoretical model is based upon the shallow water assumption and an unsteady non-Darcy flow in an isotropic porous medium. The transfer of momentum from the porous body to the interstitial fluid is effected by a Darcy-Forchheimer type interaction, thus introducing two free parameters which must be determined through matching with experiments. A finite difference scheme with flux corrected transport is used to integrate these equations. Numerical results agree favorably with those from observations.