关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

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Numerical Approximation Of Partial Differential Equations

Numerical study of flow through random packing of non-overlapping spheres in a cylindrical geometry is investigated. Dimensionless pressure drop has been studied for a fluid through the porous media at moderate Reynolds numbers (based on pore permeability and interstitial fluid velocity), and numerical solution of Navier-Stokes equations in three dimensional porous packed bed illustrated in excellent agreement with those reported by Macdonald [1979] in the range of Reynolds number studied. The results compare to the previous work (Soleymani et al., 2002) show more accurate conclusion because the problem of channeling in a duct geometry. By injection of solute into the system, the dispersivity over a wide range of flow rate has been investigated. It is shown that the lateral fluid dispersion coefficients can be calculated by comparing the concentration profiles of solute obtained by numerical simulations and those derived analytically by solving the macroscopic dispersion equation for the present geometry.Copyright © 2006 by ASME

Simulation of Fluid Flow Through a Granular Bed

A Treatise on Electricity and Magnetism, Vol. 1

We investigate a two-dimensional microscale model for crystal dissolution and precipitation in a porous medium. The model contains a free boundary and allows for changes in the pore volume. Using a level set formulation of the free boundary, we apply a formal homogenization procedure to obtain upscaled equations. For general microscale geometries, the homogenized model that we obtain falls in the class of distributed microstructure models. For circular initial inclusions the distributed microstructure model reduces to a system of partial differential equations coupled with an ordinary differential equation. In order to investigate how well the upscaled equations describe the behavior of the microscale model, we perform numerical computations for a test problem. The numerical simulations show that for the test problem the solution of the homogenized equations agrees very well with the averaged solution of the microscale model.

Crystal precipitation and dissolution in a porous medium : effective equations and numerical experiments

Various processes such as heterogeneous reactions or biofilm growth alter a porous medium’s underlying geometric structure. This significantly affects its hydrodynamic parameters, in particular the medium’s effective permeability. An accurate, quantitative description of the permeability is, however, essential for predictive flow and transport modeling. Well-established relations such as the Kozeny–Carman equation or power law approaches including fitting parameters relate the porous medium’s porosity to a scalar permeability coefficient. Opposed to this, upscaling methods directly enable calculating the full, potentially anisotropic, permeability tensor. As input, only the geometric information in terms of a representative elementary volume is needed. To compute the porosity–permeability relations, supplementary cell problems must be solved numerically on this volume and their solutions must be integrated. We apply this approach to provide easy-to-use quantitative porosity–permeability relations that are based on representative single grain, platy, blocky, prismatic soil structures, porous networks, and real geometries obtained from CT-data. As a discretization method, we use discontinuous Galerkin method on structured grids. To make the relations explicit, interpolation of the obtained data is used. We compare the outcome with the well-established relations and investigate the ranges of the validity. From our investigations, we conclude whether Kozeny–Carman-type or power law-type porosity–permeability relations are more reasonable for various prototypic representative elementary volumes. Finally, we investigate the impact of a microporous solid matrix onto the permeability.

Beyond Kozeny–Carman: Predicting the Permeability in Porous Media

An accurate quantitative description of the diffusion coefficient in a porous medium is essential for predictive transport modeling. Well-established relations, such as proposed by Buckingham, Penman, and Millington–Quirk, relate the scalar diffusion coefficient to the porous medium’s porosity. To capture the porous medium’s structure in more detail, further models include fitting parameters, geometric, or shape factors. Some models additionally account for the tortuosity, e.g., via Archie’s law. A validation of such models has been carried out mainly via experiments relating the proposed description to a specific class of porous media (by means of parameter fitting). Contrary to these approaches, upscaling methods directly enable calculating the full, potentially anisotropic, effective diffusion tensor without any fitting parameters. As input only the geometric information in terms of a representative elementary volume is needed. To compute the diffusion–porosity relations, supplementary cell problems must be solved numerically and their (flux) solutions must be integrated. We apply this approach to provide easy-to-use quantitative diffusion–porosity relations that are based on representative single grain, platy, blocky, prismatic soil structures, porous networks, or random porous media. As a discretization method, we use the discontinuous Galerkin method on structured grids. To make the relations explicit, interpolation of the obtained data is used. We furthermore compare the obtained diffusion–porosity relations with the well-established relations mentioned above and also with the well-known Voigt–Reiss or Hashin–Shtrikman bounds. We discuss the ranges of validity and further provide the explicit relations between the diffusion and surface area and comment on the role of a tortuosity–porosity relation.

Old and New Approaches Predicting the Diffusion in Porous Media

In the first part of this article, we extend the formal upscaling of a diffusion–precipitation model through a two-scale asymptotic expansion in a level set framework to three dimensions. We obtain upscaled partial differential equations, more precisely, a non-linear diffusion equation with effective coefficients coupled to a level set equation. As a first step, we consider a parametrization of the underlying pore geometry by a single parameter, e.g. by a generalized “radius” or the porosity. Then, the level set equation transforms to an ordinary differential equation for the parameter. For such an idealized setting, the degeneration of the diffusion tensor with respect to porosity is illustrated with numerical simulations. The second part and main objective of this article is the analytical investigation of the resulting coupled partial differential equation–ordinary differential equation model. In the case of non-degenerating coefficients, local-in-time existence of at least one strong solution is shown by applying Schauder's fixed point theorem. Additionally, non-negativity, uniqueness, and global existence or existence up to possible closure of some pores, i.e. up to the limit of degenerating coefficients, is guaranteed.

Strong solvability up to clogging of an effective diffusion–precipitation model in an evolving porous medium

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.

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

The article concerns the growth of biofilms made by chemotactical bacteria within a saturated porous media. The increase of a biomass on the surface of the solid matrix changes the porosity and impedes the flow through the pores. By formal periodic homogenization we derive an averaged model describing the process via Darcy's law and upscaled transport equations with effective coefficients given by the evolving microstructure at the pore-scale. Based on the assumption of uniform evolve of the underlying pore geometry and slight self-diffusivity of the bacteria, solvability in a weak sense global in time or at least up to a possible clogging phenomenon is shown in the two-dimensional case. Furthermore, by assuming sufficient regularity on the data we prove the boundedness of the solution.

Raphael Schulz

Papers

Beyond Kozeny–Carman: Predicting the Permeability in Porous Media

Old and New Approaches Predicting the Diffusion in Porous Media

Strong solvability up to clogging of an effective diffusion–precipitation model in an evolving porous medium

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

An Effective Model for Biofilm Growth Made by Chemotactical Bacteria in Evolving Porous Media