Homogenization of the Stokes Problem With Non-homogeneous Slip Boundary Conditions

We study the Stokes system with non-homogeneous Fourier boundary conditions depending on a parameter, in a domain with periodic inclusions of the size of the period. Following the values of this parameter, we obtain at the limit a Darcy's law, a Brinkmann type equation or a Stokes type equation. We also present a physical model to which the results apply. This model describes the flow of an incompressible viscous fluid through a porous medium under the action of an exterior electric field.     

Expansions at small Reynolds numbers for the flow past a porous circular cylinder

The purpose of this article is to use the method of matched asymptotic expansions (MMAE) in order to study the two-dimensional steady low Reynolds number flow of a viscous incompressible fluid past a porous circular cylinder. We assume that the flow inside the porous body is described by the continuity and Brinkman equations, and the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. Formal expansions for the corresponding stream functions are used. We show that the force exerted by the exterior flow on the porous cylinder admits an asymptotic expansion with respect to low Reynolds numbers, whose terms depend on the characteristics of the porous cylinder. In addition, by considering Darcy's law for the flow inside the porous circular cylinder, an asymptotic formula for the force on the cylinder is obtained. Also, a porous circular cylinder with a rigid core inside is considered with Brinkman equation inside the porous region. Stress jump condition is used at the porous–liquid interface together with the continuity of velocity components and continuity of normal stress. Some particular cases, which refer to the low Reynolds number flow past a solid circular cylinder, have also been investigated.     

The Potential Fluid Flow Problem and the Convergence Rate of the Minimal Residual Method

In the paper the potential fluic flow problem in porous media using Darcy's law and the continuity equation is solved. Mixed-hybrid finite element formulation based on general trilateral prismatic elements is considered. Spectral properties of resulting symmetric indefinite system of linear equations are examined. Minimal residual method for the solution of systems with a symmetric indefinite matrix is applied. The rate of convergence and the asymptotic convergence factor which depend on the eigenvalue distribution of the system matrix are estimated.     

Simulation of a compositional model for multiphase flow in porous media

In this article, we consider the simulation of a compositional model for three‐dimensional, three‐phase, multicomponent flow in a porous medium. This model consists of Darcy's law for volumetric flow velocities, mass conservation for hydrocarbon components, thermodynamic equilibrium for mass interchange between phases, and an equation of state for saturations. A discretization scheme based on the block‐centered finite difference method for pressures and compositions is developed. Numerical results are reported for the benchmark problem of the third comparative solution project (CSP) organized by the society of petroleum engineers (SPE). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

A Robin-Robin Domain Decomposition Method for a Stokes-Darcy Structure Interaction with a Locally Modified Mesh

A new numerical method based on locally modified Cartesian meshes is proposed for solving a coupled system of a fluid flow and a porous media flow. The fluid flow is modeled by the Stokes equations while the porous media flow is modeled by Darcy's law. The method is based on a Robin-Robin domain decomposition method with a Cartesian mesh with local modifications near the interface. Some computational examples are presented and discussed.     

Uniqueness of weak solutions for a pseudo-parabolic equation modeling two phase flow in porous media

In this paper, we prove the uniqueness of weak solutions for a pseudo-parabolic equation modeling two-phase flow in a porous medium, where dynamic effects are included in the capillary pressure. We transform the equation into an equivalent system, and then prove the uniqueness of weak solutions to the system which leads to the uniqueness of weak solutions for the original model.     

Derivation and analysis of an effective model for biofilm growth in evolving porous media

The present article deals with the growth of biofilms produced by bacteria within a saturated porous medium. Starting from the pore‐scale, the process is essentially described by attachment/detachment of mobile microorganisms to a solid surface and their ability to build biomass. The increase in biomass on the surface of the solid matrix changes the porosity and impedes flow through the pores. Using formal periodic homogenization, we derive an averaged model describing the process via Darcy's law and upscaled transport equations with effective coefficients provided by the evolving microstructure at the pore‐scale. Assuming, that the underlying pore geometry may be described by a single parameter, for example, porosity, the level set equation locating the biofilm‐liquid interface transforms into an ordinary differential equation (ODE) for the parameter. For such a setting, we state significant analytical and algebraic properties of these effective parameters. A further objective of this article is the analytical investigation of the resulting coupled PDE–ODE model. In a weak sense, unique solvability either global in time or at least up to a possible clogging phenomenon is shown. Copyright © 2016 John Wiley & Sons, Ltd.     

Some exact solutions for fractional generalized Burgers’ fluid in a porous space

This work is concerned with deriving the equation for describing the magnetohydrodynamic (MHD) flow of a fractional generalized Burgers’ fluid in a porous space. Modified Darcy's law has been taken into account. Closed form solutions for velocity are obtained in three problems. The solutions for Navier–Stokes, second grade, Maxwell, Oldroyd-B and Burgers’ fluids appear as the limiting cases of the obtained solutions. A parametric study of some physical parameters involved in the problems is performed to illustrate the influence of these parameters on the velocity profiles.     

A unified stabilized method for Stokes’ and Darcy's equations

We use the lowest possible approximation order, piecewise linear, continuous velocities and piecewise constant pressures to compute solutions to Stokes equation and Darcy's equation, applying an edge stabilization term to avoid locking. We prove that the formulation satisfies the discrete inf-sup condition, we prove an optimal a priori error estimate for both problems. The formulation is then extended to the coupled case using a Nitsche-type weak formulation allowing for different meshes in the two subdomains. Finally, we present some numerical examples verifying the theoretical predictions and showing the flexibility of the coupled approach.     

Numerical analysis of finite element methods for miscible displacements in porous media

Finite element methods are used to solve a coupled system of nonlinear partial differential equations, which models incompressible miscible displacement in porous media. Through a backward finite difference discretization in time, we define a sequentially implicit time-stepping algorithm that uncouples the system at each time-step. The Galerkin method is employed to approximate the pressure, and accurate velocity approximations are calculated via a post-processing technique involving the conservation of mass and Darcy's law. A stabilized finite element ( SUPG ) method is applied to the convection–diffusion equation delivering stable and accurate solutions. Error estimates with quasi-optimal rates of convergence are derived under suitable regularity hypotheses. Numerical results are presented confirming the predicted rates of convergence for the post-processing technique and illustrating the performance of the proposed methodology when applied to miscible displacements with adverse mobility ratios. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 519–548, 1998     

Fluid flow in an inhomogeneous granular medium

The seepage of a compressible fluid in an inhomogeneous undeformable granular medium is investigated. It is assumed that the fluid flow in a porous space is described by the Navier–Stokes equations. It is shown that, in the case of an inhomogeneous velocity field, a tensor of additional effective stresses occurs in connection with the transfer of fluid particles in a transverse direction when flow occurs around the granules of the medium in a longitudinal direction. Using the fundamental propositions of Reynolds’ averaging theory and Prandtl's mixing path, the structure of the effective viscosity coefficient is determined and hypotheses are formulated which enable it to be assumed to be independent of the flow velocity. It is established by comparison with experimental data that the effective viscosity coefficient can exceed the viscosity coefficient of the flowing fluid by an order of magnitude. The equations of average motion are obtained, which in the case of an incompressible fluid have the form of the Navier–Stokes equations with body forces proportional to the velocity. It is established that, in addition to the well-known dimensionless flow numbers, there is a new number which characterizes the ratio of the Darcy porous drag forces to the effective viscosity forces. The proposed equations are extended to the case of the flow of an aerated fluid. The components of the angular momentum vector are used as the required functions instead of the components of the velocity vector. This enables a solving system of equations to be obtained, which, apart from the notation, is identical with the similar equations for the case of an incompressible fluid. The solution of a new problem of the fluid flow in a plane channel with permeable walls is presented using three models: Darcy's law for an incompressible and aerated fluid, and also of an aerated fluid taking the effective viscosity into account. It is established that, for the same pressure drop, the maximum flow rate corresponds to Darcy's law. Compressibility leads to its reduction, but by simultaneously taking into account the compressibility and the effective viscosity one obtains minimum values of the flow rate. The effective viscosity and aeration of the fluid has a considerable effect on the flow parameters.     

On the well-posedness of incompressible flow in porous media with supercritical diffusion

In this article we study the heat transfer equation with a supercritical diffusion term of an incompressible fluid in porous media governed by Darcy's law. We obtain the global well-posedness for small initial data belonging to critical Besov spaces and the local well-posedness for arbitrary initial data. We further show the pointwise blowup criterion.     

On the Boundary Condition at the Surface of a Porous Medium

A theoretical justification is given for an empirical boundary condition proposed by Beavers and Joseph [1]. The method consists of first using a statistical approach to extend Darcy's law to non homogeneous porous medium. The limiting case of a step function distribution of permeability and porosity is then examined by boundary layer techniques, and shown to give the required boundary condition. In an Appendix, the statistical approach is checked by using it to derive Einstein's law for the viscosity of dilute suspensions.     

On existence and uniqueness for a coupled system modeling immiscible flow through a porous medium

We consider a system of nonlinear coupled partial differential equations that models immiscible two-phase flow through a porous medium. A primary difficulty with this problem is its degenerate nature. Under reasonable assumptions on the data, and for appropriate boundary and initial conditions, we prove the existence of a weak solution to the problem, in a certain sense, using a compactness argument. This is accomplished by regularizing the problem and proving that the regularized problem has a unique solution which is bounded independently of the regularization parameter. We also establish a priori estimates for uniqueness of a solution.     

The smoothing property for a class of doubly nonlinear parabolic equations

We consider a class of doubly nonlinear parabolic equations used in modeling free boundaries with a finite speed of propagation. We prove that nonnegative weak solutions satisfy a smoothing property; this is a well-known feature in some particular cases such as the porous medium equation or the parabolic -Laplace equation. The result is obtained via regularization and a comparison theorem.

     

The stationary Stefan problem with convection governed by a non‐linear Darcy's law

We consider the bidimensional stationary Stefan problem with convection. The problem is governed by a coupled system involving a non‐linear Darcy's law and the energy balance equation with second member in L¹. We prove existence of at least one weak solution of the problem, using the penalty method and the Schauder fixed point principle. Copyright © 1999 John Wiley & Sons, Ltd.     

On Two‐Phasic Flow Problems in Elasto‐Plastic Porous Materials

In the present contribution, the formulation of the governing equations of coupled flow and deformation processes in porous materials is based on the well‐founded Theory of Porous Media (TPM) [2, 3]. Embedded in this concept, the model under consideration represents a triphasic medium of a cohesive‐frictional elasto‐plastic solid skeleton and a binary pore‐fluid, which is composed of a materially incompressible wetting phase (here water) and a materially compressible non‐wetting phase (here air). The unsaturated domain (saturation in terms of liquid saturation) of the porous medium is included in the model by the application of a suitable capillary‐pressure‐saturation relation, which takes into account the interaction of the solid skeleton and the two pore‐fluids. Furthermore, the interaction is described by Darcy's filter law including a relative permeability, which depends on the deformation of the pore space and the degree of saturation.     

Adaptive finite element solution of the porous medium equation in pressure formulation

The pressure formulation of the porous medium equation has been commonly used in theoretical studies due to its much better regularities than the original formulation. The goal here is to study its use in the adaptive moving mesh finite element solution. The free boundary is traced explicitly through Darcy's law. The method is shown numerically second‐order in space and first‐order in time in the pressure variable. Moreover, the convergence order of the error in the location of the free boundary is almost second‐order in the maximum norm. However, numerical results also show that the convergence order in the original variable stays between first‐order and second‐order in L¹ norm or between 0.5th‐order and first‐order in L² norm. Nevertheless, the current method can offer some advantages over numerical methods based on the original formulation for situations with large exponents or when a more accurate location of the free boundary is desired.     

Existence of solutions to various rock types odel model of two-phase flow in porous media

We study the flow of two immiscible and incompressible fluids through a porous media c,onsisting of different rock types: capillary pressure and relative permeablities curves are different in each type of porous media. This process can be formulated as a coupled system of partial differential equations which includes an elliptic pressurevelocity equation and a nonlinear degenerated parabolic saturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous at interfaces separating different media. A change of unknown leads to a new formulation of this problem. We derive a weak form for this new problem, which is a combination of a mixed formulation for the elliptic pressure-velocity equation and a standard variational formulation for the new parabolic equation. Under some realistic assumptions, we prove the existence of weak solutions to the implicit system given by time discretization.     

Existence of a solution for two phase flow in porous media: The case that the porosity depends on the pressure

In this paper we prove the existence of a solution of a coupled system involving a two phase incompressible flow in the ground and the mechanical deformation of the porous medium where the porosity is a function of the global pressure. The model is strongly coupled and involves a nonlinear degenerate parabolic equation. In order to show the existence of a weak solution, we consider a sequence of related uniformly parabolic problems and apply the Schauder fixed point theorem to show that they possess a classical solution. We then prove the relative compactness of sequences of solutions by means of the Fréchet-Kolmogorov theorem; this yields the convergence of a subsequence to a weak solution of the parabolic system.