Incompressible Flow In Porous Media Research Articles

Incompressible flow in porous media with fractional diffusion

In this paper we study heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy's law. We show the formation of singularities with infinite energy, and for infinite energy we obtain existence and uniqueness results of strong solutions for the sub-critical and critical cases. We prove the global existence of weak solutions for different cases. Moreover, we obtain the decay of the solution in Lp, for any p ⩾ 2, and the asymptotic behaviour is shown. Finally, we prove the existence of an attractor in a weak sense and, for the sub-critical dissipative case with α ∊ (1, 2], we obtain the existence of the global attractor for the solutions in the space Hs for any s > (N/2) + 1 − α.

On the upstream mobility scheme for two-phase flow in porous media

When neglecting capillarity, two-phase incompressible flow in porous media is modelled as a scalar nonlinear hyperbolic conservation law. A change in the rock type results in a change of the flux function. Discretizing in one-dimensional with a finite volume method, we investigate two numerical fluxes, an extension of the Godunov flux and the upstream mobility flux, the latter being widely used in hydrogeology and petroleum engineering. Then, in the case of a changing rock type, one can give examples when the upstream mobility flux does not give the right answer.

Multistep characteristic method for incompressible flow in porous media

In this paper we present a multistep difference scheme for the problem of miscible displacement of incompressible fluid flow in porous media. The discretization involves a three-level time scheme based on the characteristic method and a five-point finite difference scheme for space discretization. We prove that the convergence is of order O ( h 2 + ( Δ t ) 2 ) , which is in contrast to the convergence of order O ( h + Δ t ) proved for a singlestep characteristic with the same space discretization. Numerical experiments demonstrate the stability and second-order convergence of the scheme.

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.

Global well-posedness of incompressible flow in porous media with critical diffusion in Besov spaces

In this paper we study the model of heat transfer in a porous medium with a critical diffusion. We obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in Besov spaces B ˙ p , 1 3 / p ( R 3 ) with 1 ⩽ p ⩽ ∞ by the method of modulus of continuity and Fourier localization technique.

Boundary integral equations for viscous incompressible flows in porous media or past porous bodies

AbstractThe purpose of this paper is to present the existence and uniqueness result for a boundary value problem which describes the flow of a viscous incompressible fluid past an arbitrary porous particle with a Lyapunov boundary, which is embedded in a second porous medium, by using the Brinkman model and the potential theory. We use a boundary integral method that reduces the problem to a system of second kind Fredholm integral equations that has a unique solution in some Hölder spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analytical behavior of two-dimensional incompressible flow in porous media

In this paper we study the analytic structure of a two-dimensional mass balance equation of an incompressible fluid in a porous medium given by Darcy’s law. We obtain local existence and uniqueness by the particle-trajectory method and we present different global existence criterions. These analytical results with numerical simulations are used to indicate nonformation of singularities. Moreover, blow-up results are shown in a class of solutions with infinite energy.

Renormalization Group Model of Large-Scale Turbulence in Porous Media

Renormalization group methods are used to develop a macroscopic (large-scale) turbulence model for incompressible flow in porous media. The model accounts for the large-distance and large-time behavior of velocity correlations generated by the momentum equation for a randomly stirred, incompressible flow. Utilizing the renormalization procedure, the transport equations for the large-scale modes and expressions for effective transport coefficients are obtained. Expressions for renormalized turbulent viscosity, which accounts for the ultraviolet subrange of the turbulent kinetic energy spectrum, are also obtained.

Domain-decomposition method for parallel lattice Boltzmann simulation of incompressible flow in porous media

The lattice Boltzmann method has proven to be a promising method to simulate flow in porous media. Its practical application often relies on parallel computation because of the demand for a large domain and fine grid resolution to adequately resolve pore heterogeneity. The existing domain-decomposition methods for parallel computation usually decompose a domain into a number of subdomains first and then recover the interfaces and perform the load balance. Normally, the interface recovery and the load balance have to be performed iteratively until an acceptable load balance is achieved; this costs time. In this paper we propose a cell-based domain-decomposition method for parallel lattice Boltzmann simulation of flow in porous media. Unlike the existing methods, the cell-based method performs the load balance first to divide the total number of fluid cells into a number of groups (or subdomains), in which the difference of fluid cells in each group is either 0 or 1, depending on if the total number of fluid cells is a multiple of the processor numbers; the interfaces between the subdomains are recovered at last. The cell-based method is to recover the interfaces rather than the load balance; it does not need iteration and gives an exact load balance. The performance of the proposed method is analyzed and compared using different computer systems; the results indicate that it reaches the theoretical parallel efficiency. The method is then applied to simulate flow in a three-dimensional porous medium obtained with microfocus x-ray computed tomography to calculate the permeability, and the result shows good agreement with the experimental data.

Lattice Boltzmann simulation of flow in porous media on non-uniform grids

In this paper, the Lattice Boltzmann method (LBM) is adopted to simulate the incompressible flow in porous media on non-uniform grids. The flow through porous media was simulated by including the porosity into the equilibrium distribution function and adding a force term to the evolution equation to account for the linear and non-linear drag forces of the porous medium. Besides, non-uniform grids are adopted by using the interpolation-supplemented Lattice Boltzmann equation. Numerical simulations of lid-driven cavity flow, flow in a 2D symmetric sudden expansion channel, and cavity flow in polar coordinates are carried out. The results agree well with benchmark solutions, experimental data or traditional computational fluid dynamics method solutions. The present results demonstrate the potential of the Lattice Boltzmann algorithm for numerical simulation of fluid through porous media.

Numerical solution of the Navier-Stokes equations for incompressible flow in porous media with free surface boundary

The paper presents numerical solution of the three dimensional Navier-Stokes equations for complex flow domains consisting of fluid and porous regions with free surface boundary. Flow in both regions is described by single set of conservation equations, extending the momentum equation for porous regions by additional viscous drag term according Darcy law. Free surface kinematics is “tracked” by Volume of Fluid method. Firstly the model is verified by comparision with numerical solution of Laplace equation for simple free surface flow through vertical dam. Good result agreements are observed. Application on realistic problems is presented on two cases. First case considers problem of pressure redistribution process around a deep tunnel excavation in low-conductivity porous media and second considers flow under and through porous, partially submerged bridge.

Modelling Two-Phase Incompressible Flow in Porous Media Using Mixed Hybrid and Discontinuous Finite Elements

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two numerical methods are used to discretize the equations of the two-phase flow model: mixed hybrid finite elements are used to treat the pressure equation, h-based Richards' equation and the diffusion term in the saturation equation, the advection term in the saturation equation is treated with the discontinuous finite elements. We propose a better way to calculate the nonlinear coefficients contained in our equations on each element of the discretized domain. In heterogeneous porous media, the saturation becomes discontinuous at the interface between two porous media. We show in this paper how to use the capillary pressure–saturation relationship in order to handle the saturation jump in the mixed hybrid finite element method. The two-phase flow simulator is verified against analytical solutions for some flow problems treated by other authors.

A Finite Volume Scheme for Two-Phase Immiscible Flow in Porous Media

In this paper, we prove the convergence of a numerical method for solving two-phase immiscible, incompressible flow in porous media. The method combines an upwind time implicit finite volume scheme for the saturation equation (hyperbolic-parabolic type) and a centered finite volume scheme for the Chavent global pressure equation (elliptic type). The capillary pressure is not neglected, and we study the case when the diffusion term in the saturation equation is weakly degenerated. Estimates on the approximate solution are proven; then by using compactness theorems we obtain a limit when the size of the discretization goes to zero, and we prove that this limit is the unique weak solution of the problem that we study.

Degenerate two-phase incompressible flow II: regularity, stability and stabilization

In this paper, we analyze a coupled system of highly degenerate elliptic-parabolic partial differential equations for two-phase incompressible flow in porous media. This system involves a saturation and a global pressure (or a total flow velocity). First, we show that the saturation is Hölder continuous both in space and time and the total velocity is Hölder continuous in space (uniformly in time). Applying this regularity result, we then establish the stability of the saturation and pressure with respect to initial and boundary data, from which uniqueness of the solution to the system follows. Finally, we establish a stabilization result on the asymptotic behavior of the saturation and pressure; we prove that the solution to the present system converges (in appropriate norms) to the solution of a stationary system as time goes to infinity. An example is given to show typical regularity of the saturation.

Lattice Boltzmann model for incompressible flows through porous media.

In this paper a lattice Boltzmann model is proposed for isothermal incompressible flow in porous media. The key point is to include the porosity into the equilibrium distribution, and add a force term to the evolution equation to account for the linear and nonlinear drag forces of the medium (the Darcy's term and the Forcheimer's term). Through the Chapman-Enskog procedure, the generalized Navier-Stokes equations for incompressible flow in porous media are derived from the present lattice Boltzmann model. The generalized two-dimensional Poiseuille flow, Couette flow, and lid-driven cavity flow are simulated using the present model. It is found the numerical results agree well with the analytical and/or the finite-difference solutions.

A modified form of the κ– ε model for turbulent flows of an incompressible fluid in porous media

This work extends the recently published work of Antohe and Lage on the development of a macroscopic two-equation turbulence model for an incompressible flow in porous media. The difference occurs in approximating the Forchheimer term in the time-averaged momentum equation. Unlike the Antohe and Lage’s work, where only the linear terms of the expansion are kept, in the presently proposed model we include the second-order correlation term. This inclusion gives rise to an extra term in the transport momentum equation which, in turn, gives rise to additional terms in the transport equations for the turbulent kinetic energy and the dissipation rate. The additional higher order terms produce correlation coefficients that are used to absorb any departure from the clear flow when expressing the two-equation turbulence model for incompressible flow in a porous medium.

A locally conservative Eulerian-Lagrangian numerical method and its application to nonlinear transport in porous media

Eulerian-Lagrangian and Modified Method of Characteristics (MMOC) procedures provide computationally efficient techniques for approximating the solutions of transport-dominated diffusive systems. The original MMOC fails to preserve certain integral identities satisfied by the solution of the differential system; the recently introduced variant, called the MMOCAA, preserves the global form of the identity associated with conservation of mass in petroleum reservoir simulations, but it does not preserve a localized form of this identity. Here, we introduce an Eulerian-Lagrangian method related to these MMOC procedures that guarantees conservation of mass locally for the problem of two-phase, immiscible, incompressible flow in porous media. The computational efficiencies of the older procedures are maintained. Both the original MMOC and the MMOCAA procedures for this problem are derived from a nondivergence form of the saturation equation; the new method is based on the divergence form of the equation. A reasonably extensive set of computational experiments are presented to validate the new method and to show that it produces a more detailed picture of the local behavior in waterflooding a fractally heterogeneous medium. A brief discussion of the application of the new method to miscible flow in porous media is included.

Finite element approximations of two-phase miscible incompressible displacement with discontinuous coefficients

Two-phase, miscible, incompressible flow in porous media is governed by a system of nonlinear partial differential equations. Many numerical methods have been given by different authors to this system, but these methods need very high regularity conditions. Actually, in most practical applications these regularity conditions couldn’t be satisfied. In this paper, the problem of discontinuous coefficients with lower regularity conditions is considered and the error estimates are demonstrated.

A general two-equation macroscopic turbulence model for incompressible flow in porous media

view of the practical and fundamental importance to heat and mass transfer, we present a two-equation turbulence model for incompressible flow within a fluid saturated and rigid porous medium. The derivation consists of time-averaging the general (macroscopic) transport equations and closing the model with the classical eddy diffusivity concept and the Kolmogorov-Prandtl relation. The transport equations for the turbulence kinetic energy (κ) and its dissipation rate (ε) are attained from the general momentum equations. Analysis of the κ-ε equations proves that for a small permeability medium, small enough to minimize the form drag (Forchheimer term), the effect of a porous matrix is to damp turbulence, as physically expected. For the large permeability case the analysis is inconclusive as the Forchheimer term contribution can be to enhance or to damp turbulence. In addition, the model demonstrates that the only possible solution for steady unidirectional flow is zero macroscopic turbulence kinetic energy. The implications of this conclusion are far reaching. Among them, this conclusion supports the hypothesis of having microscopic turbulence, known to exist at high speed flow, damped by the volume averaging process. Therefore, turbulence models derived directly from the general (macroscopic) equations will inevitably fail to characterize accurately turbulence induced by the porous matrix in a microscopic sense.

Adaptive Mesh Refinement and Multilevel Iteration for Flow in Porous Media

An adaptive local mesh refinement algorithm originally developed for unsteady gas dynamics by M. J. Berger is extended to incompressible flow in porous media. Multilevel iteration and domain decomposition methods are introduced to accommodate the elliptic/parabolic aspects of the flow equations. The algorithm is applied to a two-phase polymer flooding model consisting of a system of nonlinear hyperbolic mass conservation equations coupled to an elliptic pressure equation. While the various numerical methods used have been presented previously, our emphasis is on their consistent combination within the adaptive mesh refinement framework to treat important problems in porous media flow. To achieve efficient, easily maintainable code, we have exploited the features of object-oriented programming for the overall program structure and data management. Examples of algorithmic performance and computational results are provided.