The dispersive mixing resulting from complex flow in three-dimensionally heterogeneous porous media is analyzed using stochastic continuum theory. Stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three-dimensional, statistically anisotropic input covariance describing the hydraulic conductivity. With a statistically isotropic input covariance, the longitudinal macrodispersivity is convectively controlled, but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term. With an arbitrarily oriented anisotropic conductivity covariance, all components of the macrodispersivity tensor are convectively controlled, and the ratio of transverse to longitudinal dispersivity is of the order of 10−1. In this case the off-diagonal components of the dispersivity tensor are significant, being numerically larger than the diagonal transverse terms, and the transverse dispersion process can be highly anisotropic. Dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and Monte Carlo simulations. The theory, which treats the asymptotic condition of large displacement, indicates that a classical gradient transport (Fickian) relationship is valid for large-scale displacements.

Three‐dimensional stochastic analysis of macrodispersion in aquifers

The paper provides an overview of the challenges and progress associated with the task of numerically predicting particle-laden turbulent flows The review covers the mathematical methods based on turbulence closure models as well as direct numerical simulation (DNS) In addition, the statistical (pdf) approach in deriving the dispersed-phase transport equations is discussed The review is restricted to incompressible, isothermal flows without phase change or particle-particle collision Suggestions are made for improving closure modelling of some important correlations

On predicting particle-laden turbulent flows

The most realistic representation of a naturally occurring porous medium is a stochastic set of macroscopic elements in which the values of the three basic hydrogeologic parameters (hydraulic conductivity K, compressibility α, and porosity n) are defined by frequency distributions. A homogeneous formation under this representation is one in which the frequency distributions do not change through space. All soils and geologic formations, even the ones that are homogeneous, show random variations in the values of the hydrogeological parameters through space; that is, they are nonuniform, and a measure of the nonuniformity is provided by the standard deviation of the frequency distributions. If K and α are log normally distributed and n is normally distributed, and if we define Y = log K and C = log α, then the parameters Y, C, and n can be generated from a multivariate normal density function with means μy, μc, and μn, standard deviations σy, σc, and σn, and correlation coefficients ρyc, ρyn, and ρcn The analysis of groundwater flow in nonuniform media requires a stochastic-conceptual approach in which the effects of stochastic parameter distributions on predicted hydraulic heads are analyzed with the aid of a set of Monte Carlo solutions to the pertinent boundary value problems. In this study, two one-dimensional saturated flow problems are analyzed: steady state flow between two specified heads and transient consolidation of a clay layer. The primary output is the statistical distribution of hydraulic head ϕ, through space and time, as indicated by the mean values and their standard deviations Sϕ¯(x, t) Results show that the standard deviations of the input hydrogeologic parameters, particularly σy and σc, are important index properties; changes in their values lead to different responses for even when the means μy, μc, and μn are fixed. The degree of uncertainty associated with hydraulic head predictions increases as the degree of nonuniformity of the porous medium increases. For large values of σy and σc it becomes virtually impossible to obtain meaningful hydraulic head predictions. For transient flow the output distribution of hydraulic head values is almost never normal; in some cases it approaches a uniform distribution. The results of this study throw into question the validity of the hidden assumption that underlies all deterministic groundwater modeling; namely, that it is possible to select a single value for each flow parameter in a homogeneous but nonuniform medium that is somehow representative and hence define an ‘equivalent’ uniform porous medium. For transient flow there may be no way to define an equivalent medium. The fact that nine index parameters rather than three are required to describe a nonuniform geologic formation, the large uncertainties in predicted hydraulic heads for relatively simple flow problems in nonuniform soils, and the contention that there may be no simple way to define an equivalent uniform porous medium all have important implications in the development of groundwater flow theory and in its most fundamental applications.

A Stochastic-Conceptual Analysis of One-Dimensional Groundwater Flow in Nonuniform Homogeneous Media

In this paper, theoretical and experimental approaches to flow, hydrodynamic dispersion, and miscible and immiscible displacement processes in reservoir rocks are reviewed and discussed. Both macroscopically homogeneous and heterogeneous rocks are considered. The latter are characterized by large-scale spatial variations and correlations in their effective properties and include rocks that may be characterized by several distinct degrees of porosity, a well-known example of which is a fractured rock with two degrees of porosity---those of the pores and of the fractures. First, the diagenetic processes that give rise to the present reservoir rocks are discussed and a few geometrical models of such processes are described. Then, measurement and characterization of important properties, such as pore-size distribution, pore-space topology, and pore surface roughness, and morphological properties of fracture networks are discussed. It is shown that fractal and percolation concepts play important roles in the characterization of rocks, from the smallest length scale at the pore level to the largest length scales at the fracture and fault scales. Next, various structural models of homogeneous and heterogeneous rock are discussed, and theoretical and computer simulation approaches to flow, dispersion, and displacement in such systems are reviewed. Two different modeling approaches to these phenomena are compared. The first approach is based on the classical equations of transport supplemented with constitutive equations describing the transport and other important coefficients and parameters. These are called the continuum models. The second approach is based on network models of pore space and fractured rocks; it models the phenomena at the smallest scale, a pore or fracture, and then employs large-scale simulation and modern concepts of the statistical physics of disordered systems, such as scaling and universality, to obtain the macroscopic properties of the system. The fundamental roles of the interconnectivity of the rock and its wetting properties in dispersion and two-phase flows, and those of microscopic and macroscopic heterogeneities in miscible displacements are emphasized. Two important conceptual advances for modeling fractured rocks and studying flow phenomena in porous media are also discussed. The first, based on cellular automata, can in principle be used for computing macroscopic properties of flow phenomena in any porous medium, regardless of the complexity of its structure. The second, simulated annealing, borrowed from optimization processes and the statistical mechanics of spin glasses, is used for finding the optimum structure of a fractured reservoir that honors a limited amount of experimental data.

Flow phenomena in rocks : from continuum models to fractals, percolation, cellular automata, and simulated annealing

https://ris.utwente.nl/ws/files/6408590/discrete.pdf

Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidised bed: A hard-sphere approach.

Statistical hydromechanics of disperse systems Part 1. Physical background and general equations

Flow of dense suspensions

The theory of concentrated two-phase mixtures developed in the previous parts of this paper is applied to analysis of the structure of the local random motion (pseudo-turbulence) occurring in flows of suspensions of small solid spheres. Suspensions under study are assumed to be locally homogeneous in the sense that large-scale agglomerates of many particles or voids filled with the pure liquid do not arise in their flows and particles can be approximately regarded as statistically independent units. Coefficients of the particle diffusion caused by pseudo-turbulence are calculated without restrictions imposed on the value of the Reynolds number Re characterizing the fluid flow around one particle. Other pseudo-turbulent quantities (the r.m.s. pseudo-turbulent velocities of both phases, their effective pseudo-turbulent viscosities in a shear flow, etc.) are considered for small Re. In particular, a natural explanation is given to the known effect of the reduced hydraulic resistance of a fluidized bed as compared with that of a stationary particulate bed of the same porosity. Additionally, some properties of the mean motion of a suspension influenced by pseudo-turbulence are discussed brifly. By way of example, two problems are considered: stability of the upward flow of a homogeneous suspension with respect to small perturbations depending upon the vertical co-ordinate and time, and the spatial distribution of particles suspended by the upward flow of a fluid under gravity.

Yu.A. Buyevich

Papers

Statistical hydromechanics of disperse systems Part 1. Physical background and general equations

Flow of dense suspensions

Statistical hydromechanics of disperse systems. Part 3. Pseudo-turbulent structure of homogeneous suspensions

On the thermal conductivity of granular materials

Instability and unsteady processes of the bulk continuous crystallization.i, linear stability analysis