Muhammad Sahimi

Flow and transport in porous media and fractured rock : from classical methods to modern approaches

Part 1 Continuum versus discrete models: a hierarchy of heterogeneities and length scales long-range correlations, fractals and percolation continuum versus discrete models. Part 2 The equations of change: the equation of continuity the momentum equation the diffusion and convective-diffusion equations. Part 3 Fractal concepts and percolation theory: box-counting method and self-similar fractals self-affine fractals multifractal systems fractional Brownian motion and long-range correlations percolation processes a glance at history. Part 3 Diagenetic processes and formation of rock: diagenetic and metasomatic processes continuum models of diagenetic processes geometrical models of diagenetic processes in granular rock a geometrical model of carbonate rock diagenetic processes of fractured rock. Part 5 Morphology of porous media and fractured rock: porosity, specific surface area and tortuosity fluid saturation, capillary pressure and contact angle pore size distribution topological properties of porous media fractal properties of porous media porosity and pore size distribution of fractal porous media morphology of fractured rocks. Part 6 Models of porous media: models of macroscopic porous media models of pore surface roughness models of megascopic porous media interpolation schemes and conditional simulation. Part 7 Models of fractured rock: continuum approach - the multi-porosity models network models simulated annealing model synthetic fractal models mechanical fracture models. Part 8 Flow and transport in porous media: the volume-averaging method and derivation of Darcys law the Brinkman and Forchheimer equations predicting the permeability, conductivity and diffusivity fractal transport and non-local formulation of diffusion derivation of Archies law relation between permeability and electrical conductivity relation between permeability and nuclear magnetic resonance dynamic permeability. Part 9 Dispersion in porous media: the phenomenon of dispersion mechanisms of dispersion processes the convective-diffusion equation measurement of dispersion coefficients dispersion in simple systems dependence of dispersion coefficients on the Peclet number models of dispersion in macroscopic porous media long-time tails - dead-end pores versus disorder dispersion in short porous media dispersion in porous media with percolation disorder dispersion in megascopic porous media dispersion in stratified porous media. Part 10 Flow and dispersion in fractured rock: flow in a single fracture - continuum and discrete models flow in fractured rock dispersion in a single fracture dispersion in fractured rock. Part 11 Miscible displacements: factors affecting miscible displacement processes viscous fingering continuum models of miscible displacements in Hele-Shaw cells continuum models of miscible displacements in porous media. (Part contents).

Statistical and continuum models of fluid-solid reactions in porous media

In this review, we discuss past theoretical works on fluid-solid reactions in a porous medium. Such reactions are often accompanied by a continuous alteration of the pore structure of the medium, and at high conversions they exhibit percolation-type behavior, i.e. the solid matrix of the medium and/or the fluid phase lose their macroscopic connectivity. These phenomena are, therefore, characterized by a percolation threshold which is the volume or area fraction of a phase (solid or fluid) below which that phase exists only in isolated clusters or islands. Important classes of such processes are acid dissolution of a porous medium and gas—solid reactions with pore volume growth, e.g. coal gasification, and with pore closure, e.g. lime sulfation, and catalyst deactivation. These processes are characterized by continuous changes in the pore space as a result of a chemical reaction. We also consider here other processes such as the flow of fines, stable emulsions and solid particles in a porous medium which also alter the structure of the pore space, but by physical interaction of the particles and the solid surface of the pores. In this review we compare two different modelling approaches to reactions accompanied by structural changes. First we review the continuum approach, which is based on the classical equations of transport and reaction supplemented with constitutive equations describing the effect of structural changes on reaction and transport parameters. We then outline the relevant concepts, ideas and techniques of percolation theory and the statistical physics of disordered media, and review their application to the phenomena mentioned above. In particular, we emphasize the fundamental role of connectivity of the porous medium in such phenomena. Since in both approaches one needs to estimate the effective transport properties of the porous medium that is undergoing continuous change, we also review continuum and statistical methods of estimating the effective transport properties of disordered porous media.

A study by in situ techniques of the thermal evolution of the structure of a Mg–Al–CO3 layered double hydroxide

Abstract Several in situ techniques have been used to investigate the thermal evolution of the structure of a Mg–Al–CO 3 layered double hydroxide (LDH) under an inert atmosphere. Based on the results of the study, a model is proposed to describe the structural evolution of the Mg–Al–CO 3 LDH. According to this model as the temperature is increased, loosely held interlayer water is lost in the temperature range of 70–190°C, but the LDH structure still remains intact. The OH − group, likely in a Al–(OH)–Mg configuration, begins to disappear at 190°C, and is completely lost at 280°C; a gradual transformation of the LDH structure begins in the same range of temperatures. The OH − group, likely in a Mg–(OH)–Mg configuration, begins to disappear at 280°C and is completely lost at 405°C; a gradual degradation of the LDH structure is observed in the same range. Although some CO 3 2− loss is observed at lower temperatures, its substantial loss begins at 410°C, and is completed at 580°C. At these temperatures the material becomes an amorphous metastable, mixed solid oxide solution.

Dispersion in flow through porous media—I. One-phase flow

In this paper we extend our previous study (Sahimi et al., 1986, Chem. Engng Sci.41, 2103–2122) of dispersion processes in porous media occupied by two fluid phases. We report results of Monte Carlo investigations of dispersion in two-phase flow through disordered porous media represented by square and simple cubic networks of pores of random radii. The percolation theory of Heiba et al. (1982, SPE 11015, 57th Annual Fall Meeting of the Soc. Petrol. Engrs) is used to determine the statistical distribution of phases in the porespace. One of the phases is assumed to be strongly wetting on the porewall in the presence of the other phase. A pore size distribution is chosen which yields through the percolation theory of Heiba et al. network relative permeabilities that are in agreement with the available experimental data. As in one-phase flow dispersion is diffusive in the cases simulated, i.e. it can be described by the convective-diffusion equation. Longitudinal dispersivity in a given phase rises greatly as the saturation of that phase approaches residual (i.e. its percolation threshold); transverse dispersivity also increases, but more slowly. As residual saturation of a phase is neared, the backbone of the subnetwork occupied by the phase becomes increasingly tortuous, with local mazes spotted along it that are highly effective dispersers. Dispersivities are found to be phase, saturation and saturation history dependent. Some limited Monte Carlo experiments with a residence time representation of the effects of deadend paths within a phase or reversible adsorption on the pore walls demonstrate that the approach developed can be extended to study the influence of such delay mechanisms on the dispersion process.

Multiple-point geostatistical modeling based on the cross-correlation functions

An important issue in reservoir modeling is accurate generation of complex structures. The problem is difficult because the connectivity of the flow paths must be preserved. Multiple-point geostatistics is one of the most effective methods that can model the spatial patterns of geological structures, which is based on an informative geological training image that contains the variability, connectivity, and structural properties of a reservoir. Several pixel- and pattern-based methods have been developed in the past. In particular, pattern-based algorithms have become popular due to their ability for honoring the connectivity and geological features of a reservoir. But a shortcoming of such methods is that they require a massive data base, which make them highly memory- and CPU-intensive. In this paper, we propose a novel methodology for which there is no need to construct pattern data base and small data event. A new function for the similarity of the generated pattern and the training image, based on a cross-correlation (CC) function, is proposed that can be used with both categorical and continuous training images. We combine the CC function with an overlap strategy and a new approach, adaptive recursive template splitting along a raster path, in order to develop an algorithm, which we call cross-correlation simulation (CCSIM), for generation of the realizations of a reservoir with accurate conditioning and continuity. The performance of CCSIM is tested for a variety of training images. The results, when compared with those of the previous methods, indicate significant improvement in the CPU and memory requirements.

Non-linear and non-local transport processes in heterogeneous media: from long-range correlated percolation to fracture and materials breakdown

Abstract We review and discuss recent progress in modelling non-linear and non-local transport processes in heterogeneous media. The non-locality that we consider is caused by long-range correlations that either exist in the morphology of the media, or are caused by the transport processes themselves. The interplay between the non-linearity and non-locality is discussed in depth with the aim of establishing that, often non-linearity and non-locality are “two sides of the same coin”, such that one may have no meaning without the presence of the other one. First, we discuss linear and scalar, but non-local transport processes and, in particular, consider those in percolation systems with long-range correlations. It appears that there are significant differences between percolative transport processes in which the long-range correlations (or the covariance function) decrease with the distance r between two points, and those in which they increase as r does. Application of this problem to flow and transport in geological formations is discussed. We then consider linear vector percolation, one type of which, the rigidity percolation, provides an example of a non-local vector transport in heterogeneous media. Applications of vector percolation to modelling elastic properties of glasses, composite solids and rock, mechanical and viscoelastic properties of polymers, and vibrations and dynamical properties of heterogeneous materials are discussed. Non-linear and non-local scalar transport processes are discussed next, including various breakdown phenomena in disordered composites, power-law transport, piecewise linear transport characterized by a threshold, and non-linear processes that arise as a result of imposing a large external potential gradient on a heterogeneous medium. Their relevance to flow of non-Newtonian fluids in porous media, to electrical currents and dielectric breakdown in composite solids and doped polycrystalline semiconductors, and several other problems is then discussed. Finally, we discuss non-linear and non-local vector transport, the most important example of which is mechanical fracture of disordered solids. We discuss exactly solvable models of fracture, quasi-static and dynamic lattice models, molecular dynamics simulations, and continuum formulation of the problem. In all cases, the predictions of the models are compared with the relevant experimental data.

Pore network modelling of two-phase flow in porous rock: the effect of correlated Heterogeneity

Using large scale computer simulations and pore network models of porous rock, we investigate the effect of correlated heterogeneity on two-phase flow through porous media. First, we review and discuss the experimental evidence for correlated heterogeneity. We then employ the invasion percolation model of two-phase flow in porous media to study the effect of correlated heterogeneity on rate-controlled mercury porosimetry, the breakthrough and residual saturations, and the size distribution of clusters of trapped fluids that are formed during invasion of a porous medium by a fluid. For all the cases we compare the results with those for random (uncorrelated) systems, and show that the simulation results are consistent with the experimental data only if the heterogeneity of the pore space is correlated. In addition, we also describe a highly efficient algorithm for simulation of two-phase flow and invasion percolation that makes it possible to consider very large networks.

Stochastic transport in disordered systems

We develop a theory of stochastic transport in disordered media, starting from a linear master equation with random transition rates. A Green function formalism is employed to reduce the basic equation to a form suitable for the construction of a class of effective medium approximations (EMAs). The lowest order EMA, developed in detail here, corresponds to recent approximations proposed by Odagaki and Lax [Phys. Rev. B 24, 5284 (1981], Summerfield [Solid State Commun. 39, 401 (1981)], and Webman [Phys. Rev. Lett. 47, 1496 (1981)]. It yields an effective transition rate Wm which can be identified as the memory kernel of a generalized master equation, and used to define an associated continuous‐time random walk on a uniform lattice. The long‐time behavior of the mean‐squared displacement arising from an initially localized state can be found from Wm, as can diffusion constants in any case where the long‐time behavior of the system is diffusive. Detailed calculations are presented for seven lattice systems i...

Computer simulation of particle transport processes in flow through porous media

In this paper we develop a Monte Carlo computer simulation model for a class of particle transport processes in flow through a porous medium. This class of problems includes transport of macromolecules in porous media, fines migration, flow of stable emulsion, deep-bed filtration and size-exclusion chromatography. The porous medium is represented by a three-dimensional network of interconnected cylindrical pores with nonuniform (possibly fractal) surfaces. The effective radii of the pores are distributed according to an experimentally-measured pore size distribution. The paths of the particles throughout the pore space are determined rigorously, taking into account the effect of various forces that contribute to the interaction of the particles with the pore space. The model can also take into account the effect of possible pore plugging, particle deposition and macromolecular adsorption on the surface of the pores, in which case such phenomena are percolation processes and are characterized by a percolation threshold which is the volume fraction of the open pores below which the medium loses its macroscopic connectivity. When the model is applied to the problem of fines migration in flow through a porous medium, the predictions are in quantitative agreement with the available experimental data.

A percolation model of catalyst deactivation by site coverage and pore blockage

Abstract The problem of catalyst deactivation by active site poisoning and pore blockage, under globally kinetic control, is analyzed. The catalyst pore space is represented by a three-dimensional network of interconnected pores. As a result, the effect of morphological properties of the catalyst pore space, i.e., its geometry (pore size distribution) and topology (connectedness), on the deactivation process is investigated, for the first time, simultaneously. The concepts of percolation theory, a modern theory of statistical physics of disordered media, are employed to show that both single-pore and bundle of parallel pore models perform rather poorly and that the interconnectivity of the pores plays a fundamental role in the overall catalytic behavior. The extension of the model to more complicated systems is also discussed.