A hybrid numerical model for multiphase fluid flow in a deformable porous medium

In this paper, a fully coupled finite volume-finite element model for a deforming porous medium interacting with the flow of two immiscible pore fluids is presented. The basic equations describing the system are derived based on the averaging theory. Applying the standard Galerkin finite element method to solve this system of partial differential equations does not conserve mass locally. A non-conservative method may cause some accuracy and stability problems. The control volume based finite element technique that satisfies local mass conservation of the flow equations can be an appropriate alternative. Full coupling of control volume based finite element and the standard finite element techniques to solve the multiphase flow and geomechanical equilibrium equations is the main goal of this paper. The accuracy and efficiency of the method are verified by studying several examples for which analytical or numerical solutions are available. The effect of mesh orientation is investigated by simulating a benchmark water-flooding problem. A representative example is also presented to demonstrate the capability of the model to simulate the behavior in heterogeneous porous media.     

Indecomposable quasi‐characteristics scheme on pyramidal stencil and its application for numerical simulation of two‐phase flows through heterogeneous porous medium

A new high‐resolution indecomposable quasi‐characteristics scheme with monotone properties based on pyramidal stencil is considered. This scheme is based on consideration of two high‐resolution numerical schemes approximated governing equations on the pyramidal stencil with different kinds of dispersion terms approximation. Two numerical solutions obtained by these schemes are analyzed, and the final solution is chosen according to the special criterion to provide the monotone properties in regions where discontinuities of solutions could arise. This technique allows to construct the high‐order monotone solutions and keeps both the monotone properties and the high‐order approximation in regions with discontinuities of solutions. The selection criterion has a local character suitable for parallel computation. Application of the proposed technique to the solution of the time‐dependent 2D two‐phase flows through the porous media with the essentially heterogeneous properties is considered, and some numerical results are presented. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 44–55, 2002     

A numerical scheme for multi-point special boundary-value problems and application to fluid flow through porous layers

In this paper we propose a numerical scheme based on finite differences for the numerical solution of nonlinear multi-point boundary-value problems over adjacent domains. In each subdomain the solution is governed by a different equation. The solutions are required to be smooth across the interface nodes. The approach is based on using finite difference approximation of the derivatives at the interface nodes. Smoothness across the interface nodes is imposed to produce an algebraic system of nonlinear equations. A modified multi-dimensional Newton’s method is proposed for solving the nonlinear system. The accuracy of the proposed scheme is validated by examples whose exact solutions are known. The proposed scheme is applied to solve for the velocity profile of fluid flow through multilayer porous media.     

Hydromagnetic fluctuating flow of a couple stress fluid through a porous medium

The equations of a polar fluid of hydromagnetic fluctuating through a porous medium are cast into matrix form using the state space and Laplace transform techniques the resulting formlation is applied to a variety of problems. The solution to a problem of an electrically conducting polar fluid in the presence of a transverse magnetic field and to a probem for the flow between two parallel fixed plates is obtained. The inversion of the Laplace transforms, is carried out using a numerical approach. Numerical results for the velocity, angular velocity distribution and the induced magnetic field are given and illustrated graphically for each problems.     

On the stability and uniqueness of the flow of a fluid through a porous medium

In this short note, we study the stability of flows of a fluid through porous media that satisfies a generalization of Brinkman’s equation to include inertial effects. Such flows could have relevance to enhanced oil recovery and also to the flow of dense liquids through porous media. In any event, one cannot ignore the fact that flows through porous media are inherently unsteady, and thus, at least a part of the inertial term needs to be retained in many situations. We study the stability of the rest state and find it to be asymptotically stable. Next, we study the stability of a base flow and find that the flow is asymptotically stable, provided the base flow is sufficiently slow. Finally, we establish results concerning the uniqueness of the flow under appropriate conditions, and present some corresponding numerical results.     

The critical pressure for flow in a porous medium

Bingham flow in a porous medium is considered. This can be modelled by a random structure whose dimensions are large compared with the local scale. The principal term of the asymptotic form of the critical pressure at which the liquid starts to move in this limit is computed explicitly.     

Parametrization of the homogeneous ice nucleation rate for the numerical simulation of multiphase flow

Homogeneous ice nucleation in supercooled water is studied with the classical nucleation theory (CNT). The nucleated ice particles are assumed to possess physical properties of metastable cubic ice. The comparison with literature data on homogeneous ice nucleation rates shows a good agreement between CNT predictions and experimental nucleation rates. The CNT predicted nucleation rate, critical cluster radius, and critical nucleation work are parametrized for use in numerical simulations of the freezing process of supercooled water droplets. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of the inviscid incompressible fluid flow through a 2D porous medium

We consider the non-stationary incompressible Euler equations in a 2D porous medium. We suppose a periodic porous medium, with the period proportional to the characteristic pore size and with connected fluid part. The flow is subject to an external force, corresponding to an inflow. We start from an initial irrotational velocity and prove that the effective filtration velocity satisfies a transient filtration law. It has similarities with Darcy's law, but it now connects the time derivative of the filtration velocity with the pressure gradient. The viscosity does not appear in the filtration law any more and the permeability tensor is determined through auxiliary problems of decomposition type. Using the limit problem, we construct the correction for the fluid velocity and prove that -norm of the error is of order . Similarly, we estimate the difference between the fluid pressure and its correction in as .

     

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.     

Steady flow in a deformable porous medium

A nonlinear model for a steady flow in a deformable porous medium is considered. The flow is governed by the poroelasticity system consisting of an elasticity equation for the displacement of the porous medium and Darcy's equation for the pressure in the fluid. This poroelasticity system is nonlinear when the permeability in Darcy's equation is assumed to depend on the dilatation of the porous medium. Existence and uniqueness of a weak solution of this poroelasticity system is established under rather weak assumptions on the regularity of the data. Convergence of a finite element approximation is proved and verified through numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Effective medium model for flow through beds of porous cylindrical fibres

We have used effective medium model for beds of circular cylindrical porous fibres in order to estimate the overall bed permeability (OBP). It is assumed that a representative circular porous cylindrical fibre is inside a fluid envelope beyond which effective medium is used. Both inside the cylindrical fibre and in the effective medium, Brinkman equation is used, however of different permeabilities and in the fluid envelope Stokes equation is used. The OBP corresponding to the porous fibres is estimated when the flow direction is perpendicular to the axis of the cylindrical fibres as well as parallel to the fibres. This in turn is used to estimate the OBP corresponding to a collection of porous cylindrical fibres that are randomly oriented. We have compared the results with some existing literature.     

A nonlinear two phase fluid flow through a porous medium in presence of a well

We study a flow of fresh and salt water in a two dimensional axially symmetric coastal aquifer with a well on the central axis. The flow is governed by a nonlinear Darcy's law. We also show the behaviour of the solution when the out flow of salt water at well goes to 0. Received May 1999     

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.     

An analytic solution of an oscillatory flow through a porous medium with radiation effect

Analytical solutions for two-dimensional oscillatory flow on free convective-radiation of an incompressible viscous fluid, through a highly porous medium bounded by an infinite vertical plate are reported. The Rosseland diffusion approximation is used to describe the radiation heat flux in the energy equation. The resulting non-linear partial differential equations were transformed into a set of ordinary differential equations using two-term series. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and non-harmonic functions. The free-stream velocity of the fluid vibrates about a mean constant value and the surface absorbs the fluid with constant velocity. Expressions for the velocity and the temperature are obtained. To know the physics of the problem analytical results are discussed with the help of graph.     

On a filtration problem through a porous medium

Summary We prove the existence of a unique solution for a free boundary problem relative to the stationary flow between two water reservoirs of different levels separated by a dam of a non-homogeneous porous medium. Entrata in Redazione il 5 maggio 1973.     

Stochastic modeling of non-equilibrium multiphase flow in porous media

Non-equilibrium phenomena arising from the pore scale dynamics can have considerable effect on the large scale dynamics of multiphase flow in porous media. In such cases, the relative permeabilities and capillary pressure curves are not just simple functions of phase saturations, rather are dynamic functions of space and time. The present work proposes a stochastic approach in which particle mobility can be modelled based on the actual conductance of fluid volume inside a pore space. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of Chemical reaction on MHD flow of a visco-elastic fluid through porous medium.

The effect of heat and mass transfer on free convective flow of a visco-elastic incompressible electrically conducting fluid past a vertical porous plate through a porous medium with time dependant oscillatory permeability and suction in the presence of a uniform transverse magnetic field, heat source and chemical reaction has been studied in this paper. The novelty of the present study is to analyze the effect of chemical reaction, time dependant fluctuative suction and permeability of the medium on a visco-elastic fluid flow. It is interesting to note that presence of sink contributes to oscillatory motion leading to flow instability. Further it is remarked that presence of heat source and low rate of thermal diffusion counteract each other in the presence of reacting species.     

Rotating disk flow and heat transfer through a porous medium of a non-Newtonian fluid with suction and injection

The steady flow of an incompressible viscous non-Newtonian fluid above an infinite rotating porous disk in a porous medium is studied with heat transfer. A uniform injection or suction is applied through the surface of the disk. Numerical solutions of the non-linear differential equations which govern the hydrodynamics and energy transfer are obtained. The effect of the porosity of the medium, the characteristics of the non-Newtonian fluid and the suction or injection velocity on the velocity and temperature distributions is considered. The inclusion of the three effects, the porosity, the non-Newtonian characteristics, and the suction or injection velocity together has shown some interesting effects.