Network Models Of Porous Media Research Articles

The effect of morphological disorder on hydrodynamic dispersion in flow through porous media

The authors study hydrodynamic dispersion in flow through a disordered porous medium. The main goals are to investigate the condition(s) under which a convective-diffusion equation (CDE) cannot describe dispersion processes and to investigate the effect of the disordered morphology of the pore space on dispersion processes. They first use simple models of porous media and study dispersion processes analytically and compare the results with the predictions of the CDE. The results show that the morphology of the porous medium can strongly affect dispersion processes. They then use a Monte Carlo simulation approach to study dispersion processes in random network models of porous media which are made of interconnected capillary tubes with distributed effective radii. A percolation network is used as a prototype of porous media with disordered topology. They show that, as the percolation threshold Xc of the network is approached, there exists an anomalous and length-dependent dispersion regime that cannot be described by the CDE. They propose a generalisation of the Gaussian distribution to describe dispersion in the anomalous regime, and confirm it by Monte Carlo simulations.

Hydrodynamic dispersion near the percolation threshold: scaling and probability densities

Hydrodynamic dispersion in random network models of porous media near the percolation threshold is investigated. This is done by studying the random walk of a particle in flow through the random network. Various scaling regimes (which depend on the Peclet number which is the ratio of diffusion and convection times) are identified, and the scaling relations for the mean-square displacement of the walk, both in the direction of macroscopic flow and in directions perpendicular to the macroscopic flow, are derived and related to those of anomalous diffusion on percolation clusters. It is shown that dispersion can give rise to superdiffusion in which the mean-square displacements of the random walk grow with time faster than linearly, while the spectral dimension of such random walks can be significantly larger than two, which is the critical dimension for diffusion on fractal systems. The author proposes a new equation for the probability density of finding the random walker at a point at a given time and discusses a method by which the probability density for first passage times of the walker can be determined.

Hydrodynamic dispersion in network models of porous media.

Etude de la dispersion longitudinale d'un traceur dynamiquement neutre. Pour un modele constitue d'un reseau aleatoire de tubes, on etablit les regles exactes du mouvement du traceur sous l'action combinee de la diffusion moleculaire et de la convection. On introduit un algorithme de «propagation de probabilite» qui permet un calcul (en principe) exact de la distribution du traceur lorsqu'il s'ecoule a travers le milieu

Two-Phase Flow in Random Network Models of Porous Media

Abstract To explore how the microscopic geometry of a pore space affects the macroscopic characteristics of fluid flow in porous media, we have used approximate solutions of the Navier-Stokes equations to calculate the flow of two fluids in random networks. The model pore space consists of an array of pores of variable radius connected to a random number of nearest neighbors by throats of variable length and radius. The various size and connectedness distributions may be arbitrarily assigned, as are the wetting characteristics of the two fluids in the pore space. The fluids are assumed to be incompressible, immiscible, Newtonian, and of equal viscosity. In the calculation, we use Stokes flow results for the motion of the individual fluids and incorporate microscopic capillary force by using the Washburn approximation. At any time, the problem is mathematically identical to a random electrical network of resistors, batteries, and diodes. From the numerical solution of the latter, we compute the fluid velocities and saturation rates of change and use a discrete timestepping procedure to follow the subsequent motion. The scale of the computation has restricted us so far to networks of hundreds of pores in two dimensions (2D). Within these limitations, we discuss the dependence of residual oil saturations and interface shapes on network geometry and flow conditions.

Reservoir Volumetric Parameters Defined by Capillary Pressure Studies

Published in Petroleum Transactions, AIME, Volume 210, 1957, pages 252–259. Abstract Volumetric reservoir analysis is dependent upon a firm relationship between porosity, connate water, and net pay. Capillary pressure data on core samples interrelate these three factors. It is shown that other reservoir problems may be resolved:whether water production is indigenous or extraneous to the oil producing interval,an oil-water contact is being approached, andwhether the oil-water contact is remotely situated. Introduction Volumetric analysis of carbonate reservoirs challenges the ingenuity of the reservoir engineer due to an absence of data on the minimum oil-bearing and oil-producing pores of a given reservoir rock. Many studies with this as the ultimate objective are available on the physical characteristics and fluid-flow behaviors of the carbonate reservoir. Some of these publications are: Hassler, Brunner and Deahl's study of the role of capillarity in oil production; Jan Law's and A. C. Bulnes statistical treatment of core analysis data; Chalkley's method for estimating specific surface areas and porosity; Walter Rose's report on porosity, reserves, and recovery; Archie's textural classification of carbonate rocks; Stewart and Spurlock's large core analysis; Purcell's mercury injection method in studying capillary pressure phenomena; Stewart, Craig, and Morse's model multiple-phase flow test in investigating the relative permeability effect; and Fatt's illuminating network model of porous media. The present study shows that the static reservoir parameters of porosity, connate water, and net productive thickness may be interrelated by utilizing a statistical arrangement of mercury capillary pressure data in addition to routine core analysis. This report summarizes the techniques that are followed to obtain these relations which are applicable to carbonate and sandstone reservoirs that do not contain clays. Capillary pressure statistical studies may be extended to include clay-bearing rocks by utilizing water capillary pressure data.

The Network Model of Porous Media

Published in Petroleum Transactions, AIME, Volume 207, 1956, pages 144–181.Capillary Pressure CharacteristicsDynamic Properties of a Single Size Tube NetworkDynamic Properties of Networks With Tube Radius Distribution Abstract This paper proposes the network of tubes as a model more closely representing real porous media than does the bundle of tubes. Capillary pressure curves are derived from network models and pore size distributions are calculated from these curves. In this way is shown the difference between the true and calculated pore size distributions when the capillary pressure curve is used to obtain pore size distribution for porous media. Introduction Despite the technological importance of the laws governing flow through porous media, many of these laws have not yet been clearly formulated. This is especially true of the laws governing multiphase flow. The static properties, such as the capillary pressure curve, are also not at present interpretable correctly in terms of the pore size distribution and other structural properties of porous media. In the absence of any well founded theoretical description of fluid flow through porous media, many empirical descriptions have been proposed. In addition to the strictly empirical flow equations, some equations have been developed rigorously from simple geometrical models of the pore spaces. These equations are only as valid as is the model used in their development. The two models used in the past, the sphere pack and the bundle of tubes, have been too simple, and as a result, the equations derived from them have failed to predict the observed properties. Agreement between theory and observation has been achieved for these models by inserting parameters of doubtful physical significance.